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NELSON'S 
i 'OMMON-8CHOOL ARITHMETIC: 



THE LOW 



I THE METEJ 



I if 

In- 



ME As 



LISH 





IN MEMORIAM 
111 FLOR1AN CAJORI 







NELSON'S; 
COMMON-SCHOOL ARITHMETIC: 

/> 

DESIGNED FOE, THE USE OF THE LOWEST AS 
WELL AS THE HIGHEST CLASSES, 

AND CONTAINING 

THE APPLICATION OF ARITHMETIC TO THE 
GENERAL PURPOSES OF LIFE, 

AND 

THE METRIC SYSTEM OF WEIGHTS AND MEASURES, 

KECENTLY ADOPTKT) BY CONGRESS. 

Bv RICHARD NELSON, 

PRESIDENT OF NELSOX'S VNTON BlTSTNESS COLLEGE, AND AUTHOR OP 
NELSON'S MERCANTILE ARITHMETIC. 



SECOIT3D lEZDITIOIbT. 




u 



CINCINNATI: 
R. W. CARROLL & CO., PUBLISHERS, 

No. 117 WEST FOURTH STREET. 

18G7. 



\ 



Entered according to Act of Congress, in the year 1867, by 
KIC1IARD NELSON, 

In the Clerk's Office of the District Court of the United States, for the Souther* 
District of Ohio. 



PREFACE. 



THIS treatise differs so materially from others of ita 
class, that the space of a preface will not suffice to give 
reasons for the changes made. The author will, therefore, 
have to content himself with stating that, in the^lassifi^ 
catjoj] and treatment of subjects, he has been guided by 
his own experience and that of the most distinguished 
educators of the country ; and in the application of the 
science to the purposes of life, his authorities have been 
practical men, who were familiar with the defects of the 
old system, and desirous that the rising generation should 
be trained with more direct reference to their probable 
future callings. 

improvements in machinery having almost super- ^ 



seded the necessity for a knowledge of arithmetic in 
mechanic arts, little has been done to adapt the treatise 
to such purpose; hence, it partakes largely of the mercan- 
tile character. 



This, the author considers, will be its highest recom- 
mendation, as the destination of most American youth ia 
business; and, especially, as every man in this great Na- 
tion of Commerce is more or less engaged in mercantile 
pursuits. 

(iii) 



iy PREFACE. 

The author acknowledges numerous acts of kindness 
and courtesy received, in the preparation of this work, 
from officers of the Government, gentlemen of the legal 
and mercantile professions, teachers in different parts of 
the country, especially from the Principals of the Pub- 
lic Schools of Cincinnati, most of whom generously and 
unreservedly tendered advice regarding topics of interest 
to their schools. To the distinguished educators, A. J. 
lliCKOFP and JOHN HANCOCK the latter the friend and 
associate of the author he is under peculiar obligations. 
The former gentleman, having been the first to adopt his 
Mercantile Arithmetic, was the most competent to suggest 
further improvements and hints for the adaptation of arith^ 
metic to common school purposes; while the latter, from 
his long experience as Principal of the First Intermediate 
School in this city, and his immediate knowledge of the 
wants of teachers, proved a valuable companion during the 
progress of the work, aiding constantly in the revision of 
the proof-sheets and tendering professional advice. 

Though little reliance has been placed on written au- 
thority, many works have been called into requisition, 
principally of a legal, scientific and educational character. 
Of these, free use has been made, especially in the de- 
partment of the tables, which, it is hoped, will be found 
to subserve the present wants of every-day life. 

liicHARD NELSON. 
CINCINNATI, August, 1SOG. 



A VOCABULARY 

OF 

TECHNICAL TERMS USED IN BUSINESS. 



ABATEMENT, or deduction, an amount taken off a bill for prompt payment, 

damages, etc. 
ACCEPTANCE, agreeing to price or terms proposed; a bill with one'8 name 

written in such a way as to bind for payment. See page 197. 
ACCOMMODATION PAPER, a bill or note used to raise money, and not to pay 

a debt. 

ACCOUNT, detailed statement of goods sold. A statement showing the indebt- 
edness of one person to another. 

ACCOUNTANT, a professional calculator ; one skilled in book-keeping. 
ACCOUNT-BOOK, a ruled book in which accounts are kept, 
ACCOUNT CURRENT, a plain statement of a running account between two 

persons. See page 194. 
ACCOUNT SALES, a detailed statement of goods sold, made by an agent to his 

principal. 

ACQUITTANCE, a written discharge ; a receipt in full for money due. 
AD VALOREM, according to value, an assessment for custom duty. 
ADVANCE, a sum of money paid before value is received. 
ADVENTURE, a doubtful speculation ; a term used in book-keeping for goods 

shipped to be sold on commission. 
ADVICE, mercantile intelligence. 

AFFIDAVIT, a declaration in writing, made on oath before a magistrate, etc. 
AGENT, one who acts for another. 
ANNUITY, a sum of money paid periodically. 
ANNUL, to make void ; to cancel. 
ANTEDATE, to date beforehand. 

APOTHECARIES' WEIGHT, the weight used in compounding medicines. 
APPRAISER, a valuator. 
ARBITRATION, reference of a controversy or dispute to persons chosen by the 

parties. 
ASSESSOR, or Surveyor, one whose duty it is to estimate the value of property 

for taxation. 

ASSETS, the funds and property of a trader or person in business. 
ASSIGNEE, one to whom an assignment is made. 

5 



G NELSON'S COMMON-SCHOOL ARITHMETIC. 

ASSIGNMENT a conditional transfer of property, making it over for safe keep- 
ing. 

ASSIGNOR, one who makes an assignment. 
ASSURANCE. See Insurance. 

AVERAGE, allowance made for loss at sea; a rule in arithmetic. 
AUDIT, to examine books, vouchers, etc. 
AUDITOR, one who inspects or examines and certifies accounts. 

B. 

BAGGAGE, the wearing apparel, trunks, etc., of a traveler. 

BALANCE, a well-known instrument for weighing ; to find the difference be- 
tween two sides of an account', also that difference. 

BALANCE OF TRADE, the difference of the money value of the produce re- 
ceived and exported. If a country receives more produce and manufac- 
tures than it ships, the balance of trade is said to be against it. 

BALANCE-SHEET, a paper containing a concise statement of a merchant's 
accounts. 

BALANCING BOOKS, the business of making a balance-sheet from the ac- 
counts in the ledger. 

BALE, a package of goods or produce. 

BANCO, a word used as a prefix to paper money of some parts of Europe. 

BANK-BOOK, the pass-book of a bank. 

BANKER, a dealer in money. 

BANK HOURS, from 9 to 3 o'clock. 

BANK NOTE, a bank-bill payable to bearer. 

BANKRUPT, one who is not able to pay his debts. 

BANKS. See page 154, 

BANK STOCK, the shares of a banking company. 

BEAR, a term used to designate a person who makes it his business to depress 
the price of stocks, in order to buy up. 

BILL OF ENTRY, a list of goods entered at the custom-house. 

BILL OF EXCHANGE. See page 190. 

BILL OF LADING, a receipt from a railroad, ship, etc., for goods entered for 
conveyance from one place to another. 

BILL-HEAD, a printed form, with name of business or address. 

BILL, or BILL OF PARCELS, a detailed account of goods sold. 

BILL OF SALE, a contract under seal for the sale of goods. 

BILLS PAYABLE, the name given by a merchant or other person to notes 
made and issued, or bills, drafts, etc., accepted by him. 

BILLS RECEIVABLE, all notes taken or given in payment, except one's own. 

BLANK CREDIT, permission given by house or person to draw money on ac- 
count. 

BON A FIDE, in good faith. 

BOND, a note or deed given with pecuniary security. 

BONDED GOODS, those for which bonds are given for the duties instead of 
money. 

BROKER, an agent or factor. See Broker, page 95. 

BROKERAGE, the percentage, commission, etc., paid to a broker for buying 
or selling. 



VOCABULARY OF TECHNICAL TERMS. 7 

BULL, a term applied to a broker or stock jobber, who interests himself to 
raise the price of stocks in the market, in order to command a high sum 
for those he holds. 

BULLION, uncoined gold and silver. 

C. 

CAPITAL, stock in trade ; the amount of assets employed by a person or com- 
pany in business. 

CAPITALIST, a man of large property or means ; one who has large sums in- 
vested in stocks. 

CAPITATION, a poll tax; a tax levied on male adults. 

CARGO, a ship's load. 

CARRIAGE, the charge made for conveying goods from one place to another. 

CARTAGE, the charge for carrying goods on a cart. 

CASE, a box for holding goods or merchandise. 

CASH, the general name for coin and bank notes ; sometimes checks and sight 
bills of exchange are called cash. 

CASH-BOOK, the book in which merchants and others enter the money paid 
out and taken in. 

CASH CREDIT, the privilege of drawing money at a bank, obtained by depos- 
iting suitable security. 

CASHIER, one who has charge of money. 

CELLARAGE, privileged charge of rooms underground. 

CERTIFICATE, testimony given in writing; a paper granting some particular 
privilege. 

CHAMBER OF COMMERCE, an association of merchants for the protection 
of trade. 

CHARTER, license from government to pursue certain kinds of business. 

CHARTER PARTY, a contract in writing between the owner and freighter of a 
vessel. 

CHATTELS, all goods and real or personal property, except real estate. 

CHECK, an order on a bank for payment on demand. 

CHECK-BOOK, a printed book of blank checks. 

CHEST, a box or package; tea and opium are packed in chests. A chest of 
opium contains 111% Ibs. ; tare allowed, \% Ibs. A chest of tea is variable. 

CIRCULAR, a printed letter of advertisement. 

CLOSING AN ACCOUNT, balancing the two sides, by placing the difference 
on the smaller side, under the name of balance or profit and loss, and 
drawing lines beneath. 

CLERK, an assistant in a store, office, etc. See page 198. 

COLLECTOR, one authorized to receive money for another. 

COMMERCE, the business of exchanging one commodity for another; buying 
and selling; mercantile business. 

COMMERCIAL, pertaining to commerce. 

COMMISSION. See Commission, page 139. 

COMPANY, a number of persons associated in business. 

COMPENSATION, remuneration or reward for injury or services. 

COMPETITION, rivalry, contention for contract or supremacy. 

CONSIDERATION, a bonus; a sum given on account for any thing. 

CONSIGN, to send goods to an agent or factor for sale. 



8 NELSON'S COMMON-SCHOOL ARITHMETIC. 

CONSIGNEE, a person \vho receives goods on trust, or to be sold on com* 

mission. 

CONSIGNMENT, goods sent to a distance for sale by an agent. 
CONSIGNOR, the person or party who consigns. 
CONSOLS, public stocks in England. 
CONSUMER, one who uses or expends goods. 
CONSUMPTION, a using up; the quantity consumed. 
CONTINGENT, a share arising from an adventure; doutbful. 
CONTRA, on the other side ; per contra, a writing on the opposite side. 
CONTRABAND GOODS, articles on which there are heavy duties, or those 

wholly prohibited by government. 

CONTRACT, an agreement between parties for a lawful consideration ; a bargain. 
CONTRACTOR, one who bargains. 

CONTRIBUTION, a joint payment of money to an undertaking. 
CONVEYANCE, a legal document, transferring land or other property from 

one person to another; the transport of goods or passengers from one placo 

to another ; a vehicle. 
COOPERAGE, money paid to a cooper, or charges made for repairs of casks, 

etc. 

COPARTNER a person engaged in partnership. 

COPYING INK, adhesive ink, prepared with gum, etc., for transferring writing. 
COPYING PRESS, an instrument for taking impressions from writing; copy- 
ing letters, etc. 

CORRESPONDENT, a letter writer ; news writer for a periodical. 
COUNTER ENTRY, a contrary entry. 
COUNTERMAND, a contrary order. 

COUNTING-HOUSE, or COUNTING-ROOM, a merchant's office. 
COUPON, that part of a bond or other instrument designed to be cut off. 
CREDIT, giving trust; goods supplied without present payment. 
CURRENCY, paper money and coin established as the circulating medium of 

a country. 

CUSTOM, a tax levied on goods imported or exported. 
CUSTOMER, a regular buyer of goods at a stated place. 
CUSTOM-HOUSE, the place appointed to receive custom. 
CUSTOM-HOUSE ENTRY, a statement made, and fees and expenses paid in 

clearing a ship. 

D. 

DAMAGE, injury inflicted or sustained. 

DAMAGED GOODS, articles of merchandise or produce which have been in- 
jured. 

DAY-BOOK, the book in which merchants record daily transactions. 

DAYS OF GRACE. See page 142. 

DEBIT, to make any thing debtor in one's books ; a charge entered 

DEBIT SIDE, the left side of a page in the ledger. 

DEBT, something due to another. 

DEBTOR, one who owes another. 

DECIMAL CURRENCY, moneys reckoned by tens, as the United States cur- 
rency. 

DEED, a legal instrument of agreement under seal. 



VOCABULARY OF TECHNICAL TERMS. 9 

DEFAULT, a failure of payment. 

DEFAULTER, one who makes away with the public funds intrusted to his 
en re. 

DEFICIT, a deficiency; something wanted 

DEPOSIT, a lodgment; a pledge or pawn; money intrusted to the care of 
others, 

DEPOSITOR, one who has money lodged in bank for safe keeping. 

DEPRECIATION, lessening in value. 

DESPATCH; to transmit, to forward goods, papers, etc. See Dispatch. 

DETERIORATION, damage done ; wear and tear sustained. 

DIRECTOR, a manager, a superintendent selected by a company or board. 

DIRECTORY, an alphabetical guide or address book to the inhabitants of a city. 

DISCOUNT, a deduction ; something thrown off the amount of a bill or note; 
the sum paid by way of interest for the advance of money at bank. 

DISCOUNT BROKER, one who loans money on notes of hand. 

DISCOUNT DAY, some banks discount only on stated days, called discount 
days. 

DISPATCH, a letter or message by telegraph. 

DISSOLUTION breaking up of a copartnership. 

DITTO, the same. 

DIVIDEND, interests on stocks ; a share of the proceeds of a joint stock spec- 
ulation. 

DOCK, a secure landing for ships ; a place for landing cargoes ; also a place to 
build or repair ships. 

DOUBLE ENTRY, a method of keeping books, which considers every busi- 
ness transaction contains both a debit and a credit. 

DRAFT, an order to pay money; a deduction from the weight of goods; a 
rough copy of a writing, etc. 

ORAW, to write an order on another for money or goods. 

DRAWEE, the person on whom the bill is drawn. 

DRAWER, the person who draws a bill. 

DRAYAGE, the charge made for goods carried on a dray. 

DRUGGIST, one who sells drugs, chemicals, paints, etc. 

DRY GOODS, a commercial name for cottons, woolens, laces, etc. In England, 
for grain, coal, etc. 

DUPLICATE, a copy ; a second article of a kind. 

DUTY, a tax on goods or merchandise. 

E. 

EFFECTS, goods, property on hand, one's possessions. 

ENDORSE, to employ to the exclusion of every thing else. See Indorse. 

ENTERPRISE, an adventure ; a projected scheme. 

ENTRY, a record made in a business book ; depositing a ship's papers ofc. 
landing. 

ENGROSS, to monopolize. 

ESTIMATE, to appraise or value ; to judge by inspection. 

EXCHANGE, giving one commodity for another; a place of meeting; per- 
centage arising from the sale of bills, etc. 

EXECUTOR, a person appointed to carry out the intentions of a testator. 

EXHIBIT, a voucher or document produced in a court of law. 



10 NELSON'S COMMON-SCHOOL ARITHMETIC. 

EXPENDITURE, a charge or disbursement; outlay for expenses. 

EXPORTER, a shipper who sends goods or produce to another country for sale. 

EXPORTS, goods sent out of a country. 

EXPRESS, a special messenger; a species of conveyance. 

F. 

FACE, the amount for which a note is drawn ; also the side on which the writ- 
ing is made. 

FAC-SIMILE, an exact copy. 

FACTOR, an agent or broker. 

FAILURE, a term for suspension of payment; breaking up of business. 

FANCY GOODS, ribbons, silks, satins, etc. 

FEE, a gratuity ; the charge of a professional man for services. 

FEE SIMPLE, a property acquired by inheritance or that owned without con- 
ditions. 

FELLOWSHIP, companionship, partnership. 

FINANCE, ready money, funds or resources. 

FINANCIER, one skilled in money matters. 

FIRE INSURANCE, security against loss from fire, obtained by the payment 
of a small fee. 

FIRE POLICY, the document received from an insurance house when goods 
are insured. 

FIRM, a copartnership, a house of business. 

FLAT, low, dull, inactive. 

FLUSH, full ; an abundance of money. 

FORESTALL, to buy up goods or produce before the regular time of sale. 

FOLIO, page. 

FORWARDER, an agent who attends to the conveyance of goods, etc. 

FORWARDING HOUSE, merchants who forward goods from one place to an- 
other. 

FREIGHT, a load ; charge made for carrying goods on ship or railroad. 

FUNDS, ready money. See Public Funds. 

FURS, preserved skins of wild animals, with fine thick hair. 

FUR TRADE, the business of dealing in furs. 

G. 

GAUGE, to measure the contents of vessels, or barrels, casks, etc. ; a measure 
or standard. 

GOODS, a general name for movables. 

GROCER, a dealer in sugar, spices, dried fruits and articles of food for the table. 

GROSS, the whole weight of merchandise and box, barrel, etc. ; 12 dozen; a 
great gross is 12 times 12 dozen. 

GUARANTEE, or WARRANTY, indemnity against loss; one who binds him 
self to see the stipulation of another performed. 

GUNNY BAGS, coarse sacking made in India, used for holding coffee, rice, etc. 

H. 

HAND, a measure of four inches, used for taking the height of horses. 

HARDWARE, goods manufactured from iron. 

HAWKER, a peddler. 

HOGSHEAD, a large cask, formerly a measure of capacity. 

HORSE REPOSITORY, a place kept for the sale of horses. 



VOCABULARY OF TECHNICAL TERMS. 11 

HUNDRED WEIGHT, a hundred pounds. In England, 112 pounds 
HONOR, to accept a draft, by paying or promising to pay. 
HYPOTHECATE, to pledge as security. 

I. 

IMMOVABLES, lands, houses, fixtures, etc. 

IMMUNITY, freedom from tax, office or obligation. 

IMPERATIVE, positive, commanding. 

IMPERISHABLE, not subject to decay or waste. 

IMPORTED, brought from a foreign country. 

IMPORTER, one who brings goods from abroad. 

INCOME, receipts, gains from labor, trade, etc. 

INCONVERTIBLE, not transmissible ; ftmds that can not be converted 

stocks. 

INDORSE, to write one's name on the back of a note or draft. 
INDORSEMENT, a writing on the back of a note or other paper 
INDORSER, one who makes an indorsement. 
INITIALS, the first or capital letters of a name. 
INLAND BILLS. See page 196. 
INTEREST, right or share in business. See page 142. 
INSOLVENT, want of ability to pay. 
INSURANCE. See page 244. 

INTELLIGENCE-OFFICE, a registry office for domestics looking for situa- 
tions. 

INVENTORY, a list of goods or effects. 
INVESTMENT, capital employed ; money at interest. 
INVOICE. See page 100. 
INVOICE BOOK, a book containing invoices or copies of invoices. 

J. 

JOINT-STOCK COMPANY, an association of men to carry on heavy under- 
takings. 
JOURNAL, an intermediate book between a day-book and a ledger. 

L. 

LAND-WARRANT, a title to a lot of public land. 

LEASE, a deed ; a contract for the use of property. 

LEGAL TENDER, the authorized coins or money of a country. 

LETTER OF ADVICE, intelligence. 

LETTER OF ATTORNEY, legal authority to act for another. 

LETTER OF CREDIT, a letter from a mercantile or banking-house given to a 

traveler, by which he can collect money in a foreign country. 
LEVEE, shipping place or landing. 
LICENSE, a grant. 

LIEN, a legal claim ; power to prevent sale by another. 
LIGHTERAGE, charges for conveying goods by a lighter. 
LIQUIDATION, the act of settling debts. 

LIVE STOCK, animals kept on a farm or for sale, as cows, horses, hogs, etc. 
LLOYDS, an establishment in London for the classification of ships; a place 

of assembly for merchants and underwriters to assemble. 
LUGGAGE, baggage, the clothing, etc., of a traveler. 



12 NELSON'S COMMON-SCHOOL ARITHMETIC. 

M. 

MANIFEST, a list or exhibit of a vessel's cargo. 
MARINE, belonging or pertaining to the sea. 
MARKET, a place of sale ; price. 
MARKETABLE, what may be readily sold. 
MART, a market. 

MATERIALS, the substances from which goods and wares are made up. 
MATURITY, the time when a bill or note falls due. 

MEASUREMENT' GOODS, light goods taken on freight by measurement. 
MERCHANDISE, trade goods or wares ; goods bought to sell. 
MINT, an official place for coining money. 
MONEY BROKER, a dealer in money, bills of exchange, etc. 
MORTGAGE, a pledge of land for the payment of a debt. 
MORTGAGEE, the person who holds the pledge. 
MORTGAGER, the person who gives the pledge. 
MOVABLES, things that can be moved easily, as furniture, etc. 

N. 

NEST, a set of tubs, buckets, baskets, etc. 

NET, the clear amount, the quantity remaining after all deductions. 
NET PROCEEDS, the remainder after deducting expenses. 
NOTARIAL SEAL, the seal of a notary public. 
NOTARY PUBLIC, an officer authorized to attest documents and protest bills 

of exchange, notes, etc., for non-payment or non-acceptance. 
NOTE, a written promise to pay a debt ; a memorandum. 

O. 

OBLIGATION, a bond, a binding agreement. 
ORDER, a request to pay ; commission given to supply goods. 
ORDER-BOOK, a manufacturer's book, in which orders are copied. 

P. 

PACKAGE, a bundle. 

PACKER, a person who receives goods to pack for shipment. 
PANIC, a monetary pressure or crisis. 
PAPER, an article in common use; the name given by merchants to notes, 

bills, etc. 

PAPER CURRENCY, paper money of a country. 
PARCEL, a small package or bundle. 
PARTNER, an associate ; the member of a copartnership. 
PAR OF EXCHANGE, the value of money, both in weight and fineness, when 

compared with that of other countries. 
PASS-BOOK, a small book kept between a bank and its depositors, a merchant 

and his customers. 

PAYEE, the person to whom money is to be paid. 
PEDDLE, to carry about goods for sale. 

PERSONAL PROPERTY, money and movable goods outside of one's business. 
PETTY CASH-BOOK, a memorandum book of small receipts and expenses. 
POLICY, a writing of agreement given by insurance companies. 
POST-DATE, to date after the real time. 
POSTING, transferring from day-book, journal, etc., to the ledger. 



VOCABULARY OF TECHNICAL TERMS. 13 

POWER OF ATTORNEY, authority to act for another. 

PRICE CURRENT, a published list of market prices. 

PRIME, superior. 

PRINCIPAL, the hear! of a school or business. 

PRO-FORMA, according to form. 

PROMISSORY NOTE, an engagement in writing to pay a specified sum at a 

a stated time. 

PROSPECTUS, outline or sketch of an institution, business, book, etc. 
PROTEST, an official notice from a notary public of the non-payment of a bill, 

preparatory to legal proceedings. 
PURVEYOR, one who supplies provisions. 

Q- 

QUARTER, the fourth part of any thing; a measure of weight, 25 Ibs. ; also a 
measure of length, 9 inches. 

QUOTATIONS, current price for stock and shares, or articles of produce in the 
market. 

R. 

REBATE, discount, a reduction. 

RECEIPT, an acquittance, acknowledgement of payment. 

RECEIVER, a cashier, a person appointed to take charge of property in litiga- 
tion. 

RECEIVING HOUSE, a depot or store. 

RESOURCES, funds, assets, that which may be converted into supplies. 

RETURNS, profits or receipts in business ; accounts of goods sold by an agent. 

REMITTANCE, bills or money sent from one house to another. 

RENEWAL of a bill or note, giving a new note for a longer time ; extension of 
time on notes, etc. 

S. 

SALE, an auction ; the disposal of goods to a private bidder. 

SALVAGE, a reward claimed for saving property from loss at sea. 

SAVINGS BANKS, banks of deposit, where interest is allowed on the amount 
lodged 

SCHEDULE, an inventory of goods on parchment or paper. 

SCRIP, a receipt or acknowledgment for installments paid on stocks; a partial 
receipt, to be substituted by a receipt in full when all has been paid. 

SECRETARY, a head clerk or writer ; the recording officer of a society. 

SHIP-LETTER, a letter forwarded by private ship, instead of a packet char- 
tered for that purpose. 

SHIPPED, transmitted by sea; goods forwarded by any conveyance. 

SHIPPING CLERK, a person who attends to shipping of goods. 

SHIPMENT, the goods forwarded by railroad or steamboat; a term in double- 
entry book-keeping. 

SHOP, a work-room ; the name given to stores in England. 

SIGHT, or AT SIGHT, the time when a bill is presented to a person on whom 
it is drawn. 

SIGNATURE, the name of a person written by himself. 

SILENT PARTNER, a partner who puts in capital, but does not take an active 
pjirt in the business. 

SLEEPING PARTNER, the term used in Britain for silent partner. 



14 NELSON'S COMMON-SCHOOL ARITHMETIC. 

SMUGGLING, passing goods into a country clandestinely, so as to avoid the 

duties. 
STAPLE, the commodities which always meet with ready sale; the principal 

articles of produce or manufacture of a country. 
STERLING, according to a fixed standard ; a term applied to the money of 

Great Britain. 

STOCK, goods kept for sale ; materials of manufacture ; animals on a farm. 
STORAGE, charge for the use of a warehouse. 
STORES, supplies laid in for a ship. 
SUNDRIES, in hook-keeping, more than one; plurality. 
SUSPENDED, temporarily removed from employment; alleged inability to 

pay debts ; stoppage of work or business. 

T. 

TELLER, an officer in a bank who receives or pays money. 
TIERCE, a cask containing about 42 gallons. 

TRADE, the commerce of a country ; to exchange commodities; a bargain. 
TRADE ALLOWANCE, trade price; a discount allowed to merchants. 
TRADESMAN, a mechanic; in England, a storekeeper or retailer. 
TRAFFIC, trade, exchanging commodities. 
TRANSCRIPT, a copy. 

TRANSFER, a change of property, government funds, etc. 
TRANSHIPMENT, the act of removing from one ship to another. 
TRANSPORTATION, the conveyance of goods ; a name for a forwarding com- 
pany. 
TRANSITU, on passage ; on the way from one place to another. 

V. 

VENDOR, a seller; one who disposes of goods or property. 
VENDUE, a public sale ; an auction. 

VOUCHER, an instrument of writing ; a document produced to substantiate 
a statement of disbursements. 

W. 
WALL STREET, the street in New York City where tne principal bankers are 

located. 

WAREHOUSE, store-room ; a place for depositing goods. 
WRIT, an official notice from a law court. 

To TEACHERS?. Attention is requested to the arrangement of the answers, which 
makes the book subserve the purpose of one without answers and a key. 

In some places, as on page 44, the sum of the answers to a group of questions is 
given. Should the pupil construe his work to suit this answer, the teacher can 
detect the fraud by working auy one of the questions of the group. 

In other places, as on page 40, the answers are arranged promiscuously, with 
usually one or more than the number of questions. This prevents copying or 
working from the answer, and yet encourages the learner to study. 



MONEY, WEIGHTS AND MEASURES. 



FEDERAL MONEY consists of four kinds : gold, silver, 
nickel and paper. The smallest gold coin is of the de- 
nomination of one dollar. Other gold coins are the quar- 
ter-eagle, half-eagle, eagle and double-eagle. 

The silver coins are the dollar, half-dollar, quarter-dol- 
lar, dime, half-dime and three-cent piece. 

These are also represented by paper of the same de- 
nomination. 

The nickel coins are the one and two-cent pieces. 

The established currency of the United States consists 
of the eagle, dollar, dime and mill; but accounts are kept 
in dollars and cents only. 

TROY WEIGHT is used in the sale of gold and silver 
and at the mint for coinage: 24 grains 1 pennyweight; 
20 pennyweights=l ounce ; 12 ounces=l pound. The 
signs are gr., pwt., oz. 

The caret, when applied to gold, is only a comparative 
weight, used to indicate the proportions of pure gold and 
alloy. It is ^ part of the mass of whatever weight. 18 
carets fine is J| gold, or 18 parts gold and 6 alloy. 

COMMERCIAL WEIGHT, used in selling groceries, drugs, 
etc.: 16 ounces=l pound; 2000 pounds 1 ton. Signs, 

OZ., Us., T. 

NOTE. The ounce and pound are the principal parts of avoir- 
dupois weight in use in the United States. Iron ore and hemp are 

(15) 



16 KELSON'S COMMON-SCHOOL ARITHMETIC. 

weighed by the old standard, 112 pounds to the hundred (civt.) and 
20 hundreds, or 2240 pounds, to the ton. See Weight of a Toil 
page 17. 

MEASURES OF CAPACITY. The units of measurement 
are the gallon for liquid, and the bushel for dry measure. 
The gallon contains 58872.2 grains Troy of the standard 
pound of distilled water at 39 F., weighed in air of the 
temperature of 62, and barometer pressure 30 inches. It 
contains nearly 231 cubic inches. 

The bushel contains 543391.89 grains Troy of distilled 
water, under the above conditions, and is thus the Win- 
chester bushel of 2150.42 cubic inches. 

DRY MEASURE, used for measuring grain, fruit, etc. : 2 
pints 1 quart; 8 quarts 1 peck; 4 pecks 1 bushel. 
Signs, pt., qt.j ph., bu. 

NOTES. 1. The U. S. bushel is a cylindrical vessel, 8 inches deep 
and 18 diameter, inside, and contains 2150.42 cubic inches. 

2. By statute in Ohio, the bushel for stone coal, coke and un- 
slacked lime contains 2688 cubic inches, and the measure should be 
24 inches at the top, 20 inches at the bottom and 14.1 deep, and 
contain two bushels. 

3. The bushel of New York State contains 80 pounds of pure 
water, or 2211.84 cubic inches. 

LIQUID MEASURE,* for measuring all liquids: 4 gills= 
1 pint; 2 pints 1 quart; 4 quarts 1 gallon. 

*1. Liquid measure is the old wine measure, and has superseded 
that of beer and ale measure, both in the United States and Great 
Britain. 

2. The gill is seldom used, while barrels, tierces, etc., are gauged 
and reckoned by ? gallons. 

3. The gallon contains 231 cubic inches. 

4. A pint of water weighs 1 pound. 

5. The capacity of cisterns, vats, etc., is usually reckoned in bar- 
rels and hogsheads. 31 J gallons=l barrel; 2 barrels, or G3 gal- 
lons=l hogshead. 



MONEY, WEIGHTS AND MEASURES. 



17 



WEIGHTS OF PRODUCE PER BUSHEL, according to usage 
in Cincinnati, and as fixed by statute in Ohio: 



u 
Apples dried 


?age. Stat. 
Ibs. Ibs. 
25 25 

48 48 

34 
60 60 
20 
25 
30 
62 50 
80 
70 
30 
32 
46 
56 56 
70 70 
8 
16 
60 60 
51 
34 
50 
40 
33 32 
56 
25 

Ibs. 
.. 2240 


i 
Peaches dried 


Qsage. Stat. 
Ibs. Ib,'. 

33 33 
60 60 
24 
118 
22 
60 60 
50 
56 56 

40 
60 
62 60 
45 45 
66 56 
44 44 
14 
50 50 
14 
50 50 
00 
45 
40 
30 
60 
60 60 
77.6274 

Ibs. 

.. 2240 


Barley , 


Peas 


Barley malt, weight of 
b;i ir s included 


green 


Plaster and hair 


Beans . . ... ... 


Peanuts roasted 


Bran 


Potatoes Irish 


Bran shorts .. 


sweet 


Broom-corn 


Rye... 


Buckwheat 


Rye malt, wt. of bags 
included . . . 


Coal bituminous 


cnnnel 


Salt 


Charcoal 


Seed, clover 


Coke 


timothy 


Castor beans 


flax 


Corn, shelled 


hemp 


in ear.... 68 arid 
Hair, plasteriuf 


orchard grass.... 
Hungarian grass 
blue o*rass 


wet 


Hominy 


millet 


I-iime slacked 




Malt 


sorghurn 




Ship stuff 


Middlings 


Shorts ... 


Oats 


Turnips 


( )nions 


Wheat 


Onion sets 


Water, distilled 


WEIGHTS PER TON 

Pier Iron chill mold. . 


Iron ore....... 


Pig Iron, sand molds... 
Blooms ... . 


.. 2268 
'>464 


Hemp 


.. 2240 


HAV ... 


. 2000 


THE WEIGHT OF A 

Flour 


PINT OF 

ounces. 
. ... 14 Crnsli snorfir 


ounces. 
17 


Meal 


. 18 


Brown sugar 


.... 18 


Butter.., 


.. 15 


Loaf suecar..., 


. 19 



18 NELSON'S COMMON-SCHOOL ARITHMETIC. 

WEIGHT OF A CUBIC FOOT OF 

Ibs. I Ibs. 



Cast iron 450.55 

Wrought iron 486.65 

Steel 4S9.8 

Copper 565 

Lead 708.75 

Brass 537.75 

Tin 456 

White pine 29.56 

Loose earth or sand 95 

Common soil 1-4 

Strong soil 127 

Clay 135 

Coal 45 to 55 

Charcoal..... 18 to 18.5 



Yellow pine 33.81 

White oak 35.2 

Live oak 70 

Salt water (sea) 64.3 

Freshwater 62.5 

Air 07529 

Steam 03689 

Clay 135 

Sand 113 

Cork 15 

Tallow 59 

Brick 119 

Coke 32 

Ice 58 



23 cubic feet of sand, 18 of earth or 17 of clay make a ton. 

APOTHECARIES' FLUID MEASURE, used in compounding 
medicines : 60 minims 1 fluid drachm ; 8 fluid drachms= 
1 fluid ounce; 16 fluid ounces=l pint; 8 pints=l gal- 
lon. Signs, M.j minim; /., fluid drachm; /,., fluid 
ounce; Q, pint; Cong., gallon. 

MEASURES OF TIME. Time is divided into seconds, 
minutes, hours, days, weeks, months, years and centuries. 

60 seconds 1 minute; 60 minutes 1 hour; 24 hours 
1 day; 7 days=l week; 4 weeks 1 lunar month; 12 
calendar months=l year; 365 days=l common year; 366 
days 1 leap-year ; 365 days 5 hours 48 minutes 49.7 sec- 
onds, or 365| 1 solar year. 

A leap-year is exactly divisible by 4, and has 29 days 
in February. 1860 and 1864 were leap-years. 

The calendar months are 

1. January, 31 days. 7. July, 31 days. 

2. February, 28 " 8. August, 31 " 

3. March, 31 " 9. September, 30 " 

4. April, 30 10. October, 31 

5. May, 31 " 11. November, 30 " 
ti. June, 30 " 12. December. 31 




MONEY, WEIGHTS AND MEASURES. 19 

Commencing with January, every other month has 31 days to 
July, inclusive; and commencing with August, every other month 
has 31 days to December, inclusive. 

CIRCULAR MEASURE is divided into seconds, minutes 
and degrees. 60 seconds 1 minute; 60 
minutes 1 degree; 360 degrees 1 cir- 
cumference. 

Signs, ", seconds ; ', minutes ; , de- 
grees. 

25 31' 27", 25 deg., 31 min. 27 sec. 

LINEAR MEASURE. Long Measure is used for measur- 
ing length, breadth, depth or distance. 12 inches 1 foot; 
3 feet 1 yard ; 5^ yards 1 rod, perch or pole ; 40 rods= 
1 furlong; 8 furlongs, or 320 rods 1 mile. Signs, in.> 
inches;/!?., feet; yd., yard; ?*J., rod; fur., furlong; mi. y 
mile. 

1. In a mile there are 63360 inches, 5280 feet, 1760 yards, 320 rods. 

2. The furlong is seldom used. 

3. The 12th part of an inch is called a line. 

4. Cloth is measured by the yard and fractional parts of a yard. 

MARINE MEASURE, for measuring distances at sea: 6 
feet^l fathom; 120 fathoms=l cable length: 880 fath- 
omsI mile, called a nautical or geographical mile ; 60 
geographical or 69.77 statute miles 1 degree.* 

The speed of a ship at sea is measured by an instru- 
ment called a log-line, the knots in which correspond to 
the number of miles sailed per hour. Knot is therefore 
synonymous with mile. A ship sailing at 7 knots is mov- 
ing at the rate of 7 miles an hour. 

SQUARE OR SURFACE MEASURE. Surfaces are meas- 
ured by taking the length and breadth, by long measure, 
and multiplying them together. The length and breadth 

*The depth of the sea is measured by fathoms. 



20 NELSON'S COMMON-SCHOOL ARITHMETIC. 

of the surface of a foot are 12 inches each; hence, 12 
times 12144 square inches 1 square foot; 9 square 
feet=l square yard. 

Land Measure : 30 J square yards=l square rod ; 40 
square rods=l rood ; 4 roods, or 10 square chains 1 
acre; 640 acres=l square mile; 36 square miles=l town- 
ship. Signs, sq. yds.) sq. rds., R., A. 

1. Feet and even inches may also be used in measuring land. 

2. A square of flooring or roofing is 100 square feet. 

3. A square mile is sometimes called a section. 

4. A square rood contains 272 j square feet; an acre, 48560 square 
feet. 

CUBIC OR SOLID MEASURE includes three dimensions, 
length, breadth (or width) and thickness (or depth) mul- 
tiplied together. 1728 cubic inches 1 cube foot; 27 
cubic feet 1 cubic yard. 

STONE MEASURE is applied to masonry, which is some- 
times paid for by the foot, but usually by the perch. 
24f or 25 cubic feet=l perch ; the former for private, 
the latter for public contracts, as railroad or government 
work. 

Wood Measure: Wood is sold by the cord, which should 
measure 128 cubic feet closely piled, or 138 feet if stowed 
in a boat or barge. A pile of wood 8 feet long, 4 feet 
wide and 4 feet thick contains a cord. 

BRICKLAYERS' MEASURE. The common dimensions of 
a brick are 8 inches long, 4 inches broad and 2 inches 
thick. There are 21 bricks in a cubic foot of wall, in- 
cluding mortar. 

A wall 8 inches or 1 brick in thickness contains 14 bricks to the 
square foot of surface. 

A wall 12 inches or 1 J bricks in thickness contains 21 bricks to 
the square foot of surface. 

A wall 16 inches or 2 bricks in thickness contains 28 bricks to 
the square foot of surface. 



MONEY, WEIGHTS AND MEASURES. 21 

A fall of T *0 of an inch in a mile will produce a current 
in rivers. 

Ice 2 inches thick will bear infantry ; 4 inches, cavalry 
or light guns; 6 inches, heavy field pieces. 

PAPER. For Printers. Sizes of paper made by ma- 
chinery : 

Double imperial, 32 by 44. Super royal, 21 by 27. 

Double super royal, 27 by 42. Royal, 19 by 24, 20 by 25. 
Double medium, 23 by 26, 24 by Medium, 18J by 23J. 

37 and 25 by 38. Demy, 17 by 22. 

Royal and half, 25 by 29. Folio post, 16 by 21. 

Imperial and half, 26 by 32. Foolscap, 14 by 17. 

Imperial, 22 by 32. Crown, 15 by 20. 

A sheet folded in 2 leaves is called a folio; in 4 leaves, 
a quarto; in 8 leaves, an octavo, or Svo.; in 12 leaves, a duo- 
decimo, or 12mo.; in 18 leaves, an 18mo. ; in 24 leaves, a 
24mo. 

Stationers. 24 sheets 1 quire; 20 quires 1 ream. 

Bookbinders count from 16 to 20 sheets to the quire in 
binding account books. 

Wrapping Papers are sold by the ream and bundle; 
some reams are short count; the long count reams contain 
full quires. 

SUNDRIES : 

1 barrel of flour=196 Ibs. 3 inches=l palm. 

1 barrel of pork, etc. =200 Ibs. 4 inches=l hand. 

1 firkin of butter=55 Ibs. 9 inches=l span. 

12 articles=l dozen. 3.28 feet=l meter. 

12 dozen=l gross. 3.937 or 47 J inches=l 

144 dozen==l great gross. aune. 

20 articles=l score. 

GAS. 1.43 cubic feet of gas per hour give a light equal 
that of a candle; 1.96 feet equal 4 candles; 3 cubic feet 
equal 10 candles. 



22 NELSON'S COMMON-SCHOOL ARITHMETIC. 

HORSE POWER, in machinery, is reckoned at 33000 Ibs. 
raised 1 foot in a minute; but the ordinary work of a 
horse is 22500 Ibs. per minute for 8 hours. 

STRENGTH OF A MAN. The mean effect of the power 
of a man, unaided by a machine, is the raising 70 Ibs. 1 
foot high in a second for 10 hours a day of the power 
of the horse. 

NOTE. Two men working at a windlass, at, right angles to each 
other, can raise 70 Ibs. more easily than one man can 30 Ibs. 

A foot soldier travels 70 yards, making 90 steps, in one 
minute, common time. 

In quick time, 86 yards, making 110 steps. 

In double-quick, 109 yards, making 140 steps. 

Average weight of men, 150 Ibs. each. 

Five men can stand in a space of 1 square yard. 

A man without a load travels on a level ground 8J 
hours a day, at the rate of 3.7 miles an hour, or 31 J- miles 
a day. He can carry 111 Ibs. 11 miles in a day. 

A porter, going short distances and returning unloaded, 
can carry 135 Ibs. 7 miles a day. He can carry in a 
wheelbarrow 150 Ibs. 10 miles a day. 

HAY. 10 cubic yards of meadow hay weigh a ton. 
When the hay is taken out of old, or the lower part of 
large stacks, 8 to 9 cubic yards will make a ton. 

HILLS IN AN ACRE. 3 feet apart, there are 4840 hills 
in an acre. 

BRITISH MONEY, WEIGHTS AND MEASURES. 

In Great Britain, accounts are kept in pounds, shillings, 
pence and farthings. 4 farthings 1 penny; 12 pence 1 
shilling; 20 shillings=l pound. Signs, (farthings are 
written as fractions of a penny,) d.> pence; s., shillings; 
, pounds. ^ 



MONEY, WEIGHTS AND MEASURES. 23 

The coins are the copper half-penny and penny; silver , 
three-penny, four-penny, six-penny, shilling, half-crown 
and crown pieces; gold, the half-sovereign, sovereign and 
guinea. The value of the crown is 5 shillings; the sov- 
ereign, 20 shillings; the guinea, 21 shillings. 

THE COMMERCIAL WEIGHT is the avoirdupois, of which 
there are in use the ounce, pound, stone, quarter, hundred 
and ton. 16 draclims=l ounce; 16 ounces=l pound; 14 
pounds 1 stone; 28 pounds=l quarter; 4 quarters-! 
hundred; 20 hundred 1 ton. Signs cZr., drachms; oz., 
ounces; Z6s., pounds; qrs., quarters; cwt., hundreds; T., 
tons. 

IMPERIAL MEASURES OF CAPACITY for all liquids and 
dry goods, such as grain, potatoes, etc. : 

1 gill (gl.)=8.6648 cubic inches=5 oz. water. 

4 gillsr^l pint 34,65925 cubic inches=lj Ibs. water. 

2 pints (pts.) 1 quart=G9.3185 cubic inches=2 Ibs water. 
4 quarts (qts.)=l gallon 277.274 cubic inches=10 Ibs water. 
2 gallons (gal.) 1 peck 554.548 cubic inches=20 Ibs. water. 
4 pecks (pk.)=l bushel=r2218.192 cubic inches=80 Ibs. water. 
8 bushels (bu.)=l quarterzirl7745.536 cubic in.=640 Ibs. water. 
4 quarters (qr.)=l chaldron. 

10 quarters=l last. 

The largest measure for liquids is the gallon; the smallest for 
grain, etc., the peck. 

In London, a chaldron of coal contains 36 bushels. 

NOTE. By act of Parliament, in 1824, wine, ale and dry meas- 
ures were superseded by the imperial measures of capacity. 

Time is divided into quarterly terms not recognized in 
the United States. 

In England. 

Lady day, or 1st term Mar. 25 

Midsummer, or 2d term. ..June 24 
Michaelmas, or 3d term. ..Sept. 29 
Christmas, or 4th term ...Dec. 25 



In Scotland. 



Candlemas, or 1st term. ...Feb. 2 
Whitsunday, or 2d term. ..May 15 

Lammas, or 3d term Aug. 1 

Martinmas, or 4th term.. .Nov. 11 



Linear, square or superficial, cubic measures, etc., are 
the same in both countries. 



24 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



Articles sold by "tale," or count: 12 articles=l dozen; 
12 dozen=l gross; 5 score=:l hundred; 6 score 1 long 
hundred. 

MULTIPLICATION TABLE. 



IX 1- l 


2X 1- 2 


3X 1- 3 


4X 1= 4 


1X2- 2 


2X 2= 4 


3X 2= 6 


4X 2^ 8 


IX 3= 3 


2X 3= 6 


3X 3^ 9 


4X 3= 12 


IX 4= 4 


2X 4 = 8 


3X 4= 12 


4X 4r= 16 


IX 5- 6 


2X 5 10 


3X 5= I 5 


4X r >= 20 


IX 6^ 6 


2X 6^ 12 


3X 6= 18 


4X 6^ 24 


ix T:= 7 


2X 7 = 14 


3X 7_^ 21 


4X 7=r 28 


IX 8 = 8 


2X 8= 16 


3 X 8 = 24 


4X 8= 32 


IX 9- 9 


2X 9= 18 


3X 9 = 27 


4 X 9 36 


1 X 10 = 10 


2X10= 20 


3 X 10 30 


4X10= 40 


1x11- 11 


2X 11 22 


3X11 33 


4X11 44 


1X12- 12 


2 X 12 = 24 


3 X 12 = 36 


4 X 12 = 48 


5X 1- 5 


6X 1 


7X1- 7 


8X 1 8 


5 X ' 2 = 10 


6X 2= 12 


7X 2^ 14 


8X 2r^ 16 


5X 3= 15 


6X 3= 18 


7X 3= 21 


8X 3= 24 


5X 4= 20 


6X 4= 24 


7X 4^ 28 


8X 4= 32 


5 X 6 = 25 


6X 5= 30 


IX 5= 35 


8X 5^ 40 


5X 6= 30 


6 X 6 = 36 


?X 6r= 42 


8X 6= 48 


5X 7= 35 


6 X 7 = 42 


7X 1= 49 


8 X 7 = 56 


5X 8= 40 


6 X 8 = 48 


?X 8= 56 


8X 8= 64 


5 X 9 = 45 


6 X 9 = 54 


7X 9^r 63 


8X 9rrr 72 


5X10= 50 


6X10^= 60 


7 X 10 = 70 


8 X 10 = 80 


5 X 11 55 


6X11= 66 


7X11 77 


8XH= 88 


5 X 12 = 60 


6X12= 72 


7X12:= 84 


8 X 12 = 96 


9X 1- 9 


10 x i 10 


11 X 1- 11 


12 X 1 12 


9X 2= 18 


10 X ^ 20 


11 X 2= 22 


12 X 2 = 24 


9X 3 = 27 


10 X 3 = 30 


11 X 3= 33 


12 X 3= 36 


9 X 4 ^ 36 


10 X 4= 40 


11 X 4 44 


12 X 4= 48 


9X 5= 45 


10 X 6 = 50 


11 X 5= 55 


12 X 6= 60 


9X 6= 54 


10 X 6= 60 


11 X 6 = 66 


12 X 6=r 72 


9 X 1 = S3 


10 X 7= 70 


11 X 7= 77 


12 X 7= 84 


9X 8 = 72 


10 X 8= 80 


11 X S= 88 


12 X 8 = 96 


9X 9= 81 


10 X 9= 90 


11 X 9= 99 


12 X 9 = 108 


9X10= 90 


10 X 10 = 100 


11 X 10 = 110 


12 X 10 = 120 


9X11 99 


10 X 11 = 110 


11 X 11 121 


12X11 =132 


9 X 12 108 


10 X 12 == 120 


11 X 12 132 


12 X 12 = 144 



NELSON'S 

COMMON-SCHOOL ARITHMETIC. 



I. INTRODUCTION. 

1. Arithmetic is the art or science of computing num- 
bers. 

2. The theory of Arithmetic treats of the properties 
and relations of numbers. 

3. The practice of Arithmetic shows the application of 
numbers to business, the mechanic arts, etc. 

4. AEITHMETICAL SIGNS. 

-f-, called plus, is the sign of addition. 

= is the sign of equality. 5-}-4=9, is read, five plus 
four equals nine. 

, called Minus, is the sign of subtraction. 5 3=2, 
is read, five minus three equals two. 

-r- is the sign of division. 9-r-3=3, is read, nine divided 
Ly three equah three. 

. is the decimal sign. Placed at the left of a number, 
it represents tenths, hundredths, etc. 

: : : : is the sign of proportion. 3 : 4 : : 6 : 8, is read, 
three is to four as six is to eight. 

4. 4', 4", 4'", is read, four, four prime, four second, four 
third. f 

3 (25) 



26 NELSON'S COMMON-SCHOOL ARITHMETIC. 

I/ is called the radical sign, or sign of square root. 
1/4=2, is read the square root of four equals two. 

-ty ' . A figure inserted, as the 3, indicates the root to be 
taken; 3, the cube root; 4, the fourth root. 

3 2 . A small figure written as the 2 in the margin, indi- 
cates that the number is to be raised to a corresponding 
power; 2, the second power; 3, the third power. This 
figure is called the index of the power. 



II. NOTATION AND NUMERATION. 

5. Notation is the art of representing numbers by sym- 
bols, called figures or digits. There are ten of these 
figures : 

0123456789 

naught, one, two, three, four, five, six, seven, eight, nine. 

The first is also called zero, or cipher. 

6. When a larger number than nine is to be repre- 
sented, two or more figures are used. 

7. Numeration is the method of reading these figures 
when arranged to represent numbers. For this purpose 
they are usually divided into periods of three from the 
right. 

8. The first period on the right contains units, tens and 
hundreds, thus : 

125 

bund., tens, units, 

which is read, one hundred and twenty-five. 

The second period contains units, tens and hundreds 
of thousands, thus : 

125 , 000 

thous., hund., 

which is read, one hundred and twenty -five thousand. 



\ 



NOTATION AND NUMERATION. 27 

The third period contains units, tens, and hundreds of 
millions, thus : 

125,000,000. 

mills., thous., hund., 

which is read, one hundred and twenty-five millions. 

The fourth period contains billions, the fifth trilli)ns y 
the sixth quadrillions, the seventh quintillions. 

RECAPITULATION. The first period is hundreds, the second thou- 
Bands, the third millions, the fourth billions, etc. 

To read 12376421007, we point off thus: 

12,376,421,097. 

Here are four periods the fourth is billions. The 
number is 12 billions, 376 millions, 421 thousand and 97. 

*WRITE THE FOLLOWING IN WORDS: 



1 


2 


3 


4 


15 


127027 


1464780 


27100101 


25 


184194 


1700700 


198140197 


125 


710107 


4001001 


100001009 


1125 


411001 


7119031 


600100100 


23125 


100020 


1000010 


190999009 






5 




3214100967831 


4191006848219 



WRITE THE FOLLOWING IN FIGURES: 

1. Ten. 

2. Seventeen. 

3. One hundred and twenty. 

4. Three hundred and twenty-four. 

5. One thousand and eighty. 

6. One hundred and twenty thousand five hundred. 

7. Three hundred and ninety-seven thousand four hun- 

dred and forty-four. 

* Exercises under articles 10, 11, and 12 may precede these. 



28 NELSON'S COMMON-SCHOOL ARITHMETIC. 

8. Twelve millions one hundred and twenty-five thou- 

sand one hundred and one. 

9. One billion one hundred and one thousand and one. 

10. Thirty millions eighty-five thousand one hundred 

and seven. 

11. Seventy-six trillions five hundred and forty billions 

one hundred and ten millions and sixty -seven. 

12. Two hundred millions. 

13. One hundred trillions. 

14. Seven hundred billions three thousand and seven. 

15. One thousand. 

16. One million. 

17. One trillion one hundred thousand. 

18. One billion. 

19. Thirteen billions three millions seven thousand. 

20. One hundred and ten millions and eighty-seven. 

21. Seventy-five millions six thousand and nine. 

9. FEDERAL MONEY. 

The name usually given to the money of the United 
States is Federal Money. It is reckoned by tens and hun- 
dreds. Though there are various kinds of gold, silver 
and nickel coins, money is always reckoned in dollars and 
cents, or dollars, cents and mills. 

$ is the dollar sign. 

c, the sign for cents. 

m, the sign for mills. 

$3456.87,5, is read, three thousand four hundred and fifty- 
six dollars, eighty-seven cents, five mills. 

Mills are written one place to the right of cents. In 
this book mills will sometimes be separated from cents by 
a comma, as above. 

A period is used to separate cents from dollars, the 
two first figures on the right being cents and tens of cents. 



NOTATION AND NUMERATION. 29 

WHITE THE FOLLOWING IN WORDS ! 
12 3 

$457. $31412.875 $507.974 

$25.25 $167.476 $1134.643 

$364.61 $365.323 $216.132 

$112.57 $4767.126 $345.815 

$3146.87 $116.254 $341.664 

$213.18 $67.141 $416.304 

$456.45 .75 .17c 

$1719.97 .353 .413m 

$7.90 $1.273 .674m 

$304.02 $35.144 $100.603 

$117.44 $414.124 $210.301 

$32.32 .67 c .164m 

When there are no tens of cents, a cipher is written in 
the ten's place, as in 304.02, Ex. 1, which is read three 
hundred mid four dollars and two cents. 

Separate the dollars, cents and mills in the following, 
observing to write dollars under dollars and cents under 
cents : 



4 


5 


6 


7 


307 c 


135006 m 


3171 c 


24136 c 


M 064m 


2703 m 


1463 m 


7314 m 


32002 m 


753m 


12195m 


21364 c 


100011m 


35001 m 


3697 c 


71836 m 


1826063m 


61422 c 


2468 m 


2173 c 


116003m 


71922c 


14193 c 


1897m 


31 463 c 


143684 m 


21314 c 


1367841 m 


6897 m 


900013 c 


2136821 m 


146001 c 


18987 m 


41362 c 


21367m 


14367 c 



31464 c 13687m 98076 c 314683m 

8. 3141 c, 1386 c, 19301 c, 71432 c, 7321 c, 906301 c, 
2136 c, 4563 c, 10001 c, 30351 c, 9878 c, 45121 c, 31645 c, 
13621 c, 21001 c, 60006 c, 213621 c, 113600 c. 



30 NELSON'S COMMON-SCHOOL ARITHMETIC. 

9. 21360 m, 31461 m, 14565 in, 24798 m, 46503 m, 106 m, 
796543m, 45631m, 245m, 14013m, 41634m, 14563m, 
21364 m, 7856 m, 24130 m, 45103 m, 1467 m, 4136 m. 

10. 20135m, 314102m, 14163m, 213102m, 31453m, 
12063 m, 14673 m, 24135 m, 43103 m, 156307 m, 14007 m, 
34617m, 21368m, 146m, 178936m, 213641m, 14567m, 
36845m, 213146m, 145163m. 

11. 12314m, 2136m, 100135m, 21364m, 146345m, 
213984 m, 136453 m, 1467 m, 13645 m, 14107 in, 30674 m, 
1451 in, 147431 m, 14345 m, 41367 m, 1456 m, 21314 m, 
16789m, 13674m, 14683m, 1413m, 31463m, 16703m, 
4103 m, 163 m. 

12. 1456 m, 31463 m, 21456 m, 14637 m, 2136 m, 415 in, 
1467m, 23460m, 1454m, 3417m, 14567m, 14674 in, 
13654m, 14345m, 41631m, 14367m, 1456m, 145634 in, 
1456 m, 14567 in, 4453 m, 14639 m. 

13.23145m, 12346 in, 16734m, 2146m, 1678m, 
1463m, 14596m, 3045m, 163201m, 146324m, 14567m, 
14631 m, 21393 m, 7894 m, 14637 m, 21364 in, 14567 m, 
176301 m, 21367 m, 140601 m, 14563 m. 

14. Write in columns, as before, the following: Seventy- 
five dollars, eighty-seven cents; thirty-three dollars, sixty- 
one cents, five mills; seven hundred and ninety-six dollars 
and sixty cents; five thousand dollars; five thousand three 
hundred and eighteen dollars and sixty-three cents; two 
hundred and fifty -six dollars, fourteen cents, four mills; 
one hundred dollars, sixty cents, three mills. 

15. Thirty-six hundred dollars, seven cents, and five 
mills; eight hundred thousand dollars, forty cents, seven 
mills; sixty-seven dollars, eighty cents; nine hundred dol- 
lars, seventy-five cents, six mills; three hundred and sev- 
enty-six dollars, six cents, four mills; sixty-four dollars, 
eighteen cents, seven mills; fifty-nine dollars, six cents, 
three mills; eight hundred dollars, one cent, three mills. 



NOTATION AND NUMERATION. 31 

17. One thousand five hundred dollars, sixty cents, 
eight mills; three hundred and fifty dollars and six mills; 
seventy-eight dollars, eighty cents, six mills; four hundred 
fifty-seven dollars, sixty-four cents, seven mills. 

10. ODD NUMBERS .* 

The numbers 1, 3, 5, 7, and 9 are called odd numbers, 
and every number which has one of these figures in the 
unit's place, as 11, 13, 15, is also an odd number. 

1. Write in columns all the odd numbers from 1 to 
151, observing to keep units under units, tens under tens, 
and hundreds under hundreds. 

2. Write in the same way all the odd numbers from 
151 to 351. 

3. Write in the same way all the odd numbers from 
351 to 601. 

4. Write in the same way all the odd numbers from 
601 to 901. 

11. EVEN NUMBERS. 

The numbers 2, 4, 6, 8 are called even numbers, and 
every number which has or one of these as a unit figure, 
is also an even number. 

5. Write in columns, as before, all the even numbers 
from 2 to 200, inclusive. 

6. Write in the same way all even numbers from 200 
up to 500, inclusive. 

7. Write in the same way all the even numbers from 
500 to 800, inclusive. 

8. Write in the same way all the even numbers from 
800 to 1100, inclusive. 

The Teacher should follow these exercises by others, from dictation, 
until the scholars are taught to write any sum without hesitation. 

*With very young learners, these exercises should precede those 
on pages 26 and 27. 



32 NELSON'S COMMON-SCHOOL ARITHMETIC. 

12. ADDINO. 

In the preceding exercises the learner unconsciously 
added 2 every time he passed from one number to another. 
In the following exercises he will be required to add 3, 4, 5, 
etc., and unite the results in the same way. He should 
observe to write the figures in straight lines. 

9. Commencing at 1, add 3 every time until you reach 
97, thus: 1, 4, 7, 10, 13, putting the numbers under each 
<5ther. 

10. Commencing at 1, add 4 every time till you reach 
121. 

11. Commencing at 1, add 5 every time till you reach 
161. 

12. Commencing at 2, add 6 every time till you reach 
200. 

13. Commencing at 1, add 7 every time till you reach 
232. 

14. Commencing at 1, add 8 every time till you reach 
265. 

15. Commencing at 1, add 9 every time till you reach 
307. 

16. Add 3 to all the odd numbers up to 101. For this 
purpose write the odd numbers on the left 'of the new 
numbers, thus: 

1 4 
3 6 

17. Add, in the same way, 4, 5 and 6 up to 51. 

18. Add, in the sam'e way, 7, 8 arid 9 up to 51. 

19. Add 3, 4, 5 and 6 to all the ven numbers up to 50. 

20. Acid 7, 8 and 9 to all even numbers up to 80. 

21. Add, in the same way, 2 to the following numbers: 
19, 29, 39, 49, 59, 69/79, 89, 99, and add 3, 4, 5, 6 5 

8 and 9 in the same way. 



ADDITION. 33 

22. Add 2 to the following numbers, after writing them 
on the left: 8, 18, 28, 38, 48, 58, 68, 78, 88, 98. 

Add 3, 4, 5, 6, 7, 8 and 9 in the same way. 

23. Add 3, 4, 5, 6, 7, 8 and 9 to the following num- 
bers: 7, 17, 27, 37, 47, 57, 67, 77, 87 and 97. 

24. Add 4, 5, 6, 7, 8 and 9 to 6, 16, 26, 36, 46, 56, 
66, 76, 86 and 96. 

25. Add 5, 6, 7, 8 and 9 to 5, 15, 25, 35, 45, 55, 65, 75, 
85 and 95. 

26. Add 6, 7, 8 and 9 to 4, 14, 24, 34, 44, 54, 64, 74, 
84 and- 94. 

27. Add 7, 8 and 9 to 3, 13, 23, 33, 43, 53, 63, 73, 
83 and 93. 

The Teacher ought to examine his scholars on the terms, signs 
and principles of each rule. In this subject, on the difference be- 
tween notation and numeration, how many figures necessary to 
write one hundred, how many a thousand, etc. 



III. ADDITION.* 

13. The method of uniting two or more numbers into 
one is called Addition. 



*The Teacher will find exercises for evening study at the end of 
this chapter. 

Beginners should not be allowed to count on their fingers or 
talk over the process. If drilled in the use of the catch-figure by 
blackboard exercises, they will not afterward resort to any of the 
slower methods of computation. The use of the catch-figure is in 
part taught in exercises, page 32 and 83. For the purpose of drill, 
it ought to be taught as follows : "7 and 9, the unit figure is what?'' 
"6." "17 and 9?" "26." "27 and 9?" "36." "Observe that 
when 7 and 9 are added, the unit figure is 6." "47 and 9?" 
"57 and 9?" "67 and 9?" 

Classes should be exercised in this way through all the combi- 
nations found iii the exercises referred to. 



34 NELSON'S COMMON-SCHOOL ARITHMETIC. 

14. The one number is called the sum, amount, total, or 
footing. 

15. The sign is +, and is called phis. When placed 
between two numbers it indicates that they are to be 
added together. 3-f-2 5, is read, three plus two equals 
Jive. 

16. In performing operations in addition, it is neces- 
sary to write the units, tens, hundreds, etc., of the one 
number under the units, tens, hundreds, etc., of the other, 
which arranges the figures in one straight line. 

1. To add together 135, 241 and 323. 

135 EXPLANATION. Here units are placed under units, teng 

241 under tens and hundreds under hundreds. After arranging 
323 the figures thus, we commence at the right-hand column and 
add 3 to 1, which makes 4, to which add 5 and we have 9. 
Adding the tens' column in the same way, we have 9 
tens, which we write in the tens' place. 

Adding the hundreds' column in the same way, we have 6, which 
we write in the hundreds' place. 

Find the sum of each of the following groups : * 



2 


3 


4 


5 


6 


7 


8 


3131 


131 


211 


1534 


3143 


3141 


2131 


223 


453 


765 


1232 


2102 


5432 


1036 


115 


100 


23 


1002 


1413 


1426 


5812 


3799 


684 


999 













10 


11 


12 


13 


14 


15 


211 


3145 


4512 


2131 


14132 


14413 


1613 


101 


4132 


1035 


1027 


1734 


34441 


143 


65 


1712 


4241 


1720 


4113 


41104 


1233 





* Answers arranged promiscuously: 377, 9788,2989, 19979, 3768, 
9999, 6658, 8989, 3989, 8979, 4878, 89958. 



ADDITION. 35 

To add 325, 42 and 178. 325 

42 
178 

Answer, 545 

EXPLANATION. 1. Placing the numbers as directed, we proceed 
to find the amount of the first column on the right: 8, 2 and 5 are 
15; that is, 1 ten and 5 units. 

2. Writing the 5 units under the units, we add the 1 ten to the 
tens' column. 

#. This one added to the 7, 4 and 2 makes 14 tens or 1 hundred 
and 4 tens. 

4. Writing the 4 under the tens, we add the 1 hundred to the 
hundreds' column. 

5. This 1 added to the 1 aM 3 in the hundreds' column makes 5 
hundreds, which 5 we write in the hundreds' place and our work is 
done. 

4501 
17. To add 4501+3213+1007+302, we write 3213 

them thus: 1007 

302 



An*. 9023 
Add together the following numbers: 

18. 3478, 3167, 4199, 7854, 3456. An*. 22154 

19. 1417, 210, 61907, 216, 3184. An*. 66934 

20. 7894, 2176, 7, 109, 7998. An*. 18184 

21. 376 + 100+71+416+709+317. An*.. 1989 

22. 1006+3009+79999+7098+17. Am. 91129 

23. 316+10069+9777+307+198. An*. 20667 

24. 789632+4+67+879002+876+970 is how much? 

Ans. 1670551 

25. 98632+76398+832+97+10029+97384 is how 
much? An*. 283372 

26. 1324+4354653+12+876+97843+68473 is how 
much? Am. 4523181 



36 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



27. 31465+2316532+107+3790+465321+3654563+ 

107653+23650+1007+30672+503102+21063 is how 
much ? 

28.18230+476+41034+9875+65432+5678+12090+ 
9387+8276+565+13654+443^how much? 

Ans. Sum of 27 and 28, 7344065. 

29. Add together 45679+9837+18708+7967+485+ 
78963+84989+12345+7069+8090+7483+96748. 



TAKING TWO AND THREE FIGURES AT A TIME. 

To enable scholars to grasp two and three figures at a time, and 
carry them up as*one, they might be exercised on the blackboard 
in such sums as the following: 



136 
964 



4 3 

9 8 



4 6 

8 2 
1 3 



7 3 
1 6 
4 1 



9 2 

3 8 

4 1 



1 3 
4 5 
3 6 



693 
673 
212 



Such exercises ought to be of frequent occurrence and scholars 
encouraged to answer in concert. 

The answers should be given instantaneously, naming only the 
unit figure, as shown in the column below: 

After writing on the right of the first column the 
figures produced by pairing, the teacher may lead the 
class in adding, thus: 17 and 3? 30 and 1? 41 and 
6? 47 and 7? 54 and 1? 65 and 6? 81 and 5? 96 
and 2? 108 and 11? 

It will be observed that the tens produced iu forming 
the pairs were not named. The same course should be 
pursued in the class, as the learner is unconscious of 
making as great an effort as he really does. 

When the ten is omitted by mistake, attention should 
n be called to it by giving the full number, as 15 or 11 

instead of 5 or 1. 

1 The other columns should be added without the aid 

of the marginal figures. 

After thorough drill in this, the class should be 
taught to take three figures and even four as rapidly 
as oae. 



3456 \ 1 
1345 / L 
3689 ^ 9 
1563 f* 
9456 \ - 
3689 j D 
8998 | A 
1898 f 
9873 ) , 
1678 f 
1684 > 7 
7893 f ' 
1453 1 
1763 / 
2195 
9876 
7897 \ 
2536 / 
S529 \ - 
1438 / f 



ADDITION. 37 

30. Find the sum of 8934, 16749, 809, 67549, 98697, 
746839, 1498, 829555, 9218967, 8347912, 968000, 74685. 
Total of the preceding two, 20758557. 

Foot up the following columns : 
31 32 33 34 35 

31645 3454 4213 1565 3654 

98760 2136 631*4 3657 1095 

36875 1364 2316 5437 9014 

57893 4633 1369 3457 6789 

14567 9897 9306 1234 9687 

34564 7879 6039 3421 5764 

46387 2164 8109 6789 1567 

93178 4163 9876 1746 9139 

78163 4569 6789 3456 1456 ' 

64518 5496 4567 1378 2345 

17514 6428 5679 5932 5432 

45678 8297 3263 4567 6542 

21364 9287 9457 1657 1395 

7198 7928 1459 6574 3642 

3165 9872 1455 5638 1365 

4124 8729 9375 4932 2315 

1345 9314 5976 1397 9365 

3146 3162 7639 9765 3510 

4165 2136 7938 3765 1096 

3216 9364 3959 1456 3765 

36. Add together the following numbers: 313, 2109, 
6785, 2736, 798, 987, 21363, 316, 4934, 2178, 1009, 396, 
298, 2753, 607, 3145, 213, 6709, 6093, 190, 2130, 2160, 
716, 213, 9876, 45678, 2137, 2198, 9039, 6789, 3097, 
4684, 2136, 2178, 5672, 1987, 6789. 

Answers promiscuously arranged: 95368, 77823, 120272, 
115098, 667465, 88937, 171411. 

The Teacher should not permit his scholars to divide these col- 
umns when adding, nor should he allow them to resort to the aid 
of strokes or practice counting on their fingers. 



38 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



37 


38 


39 


40 


41 


3286 


2467 


34564 


46321 


3614 


6713 


109 


12345 


13632 


1364 


3654 


3178 


65435 


14567 


5436 


176 


145 


87654 


53678 


7835 


3976 


6178 


34564 


86367 


4678 


6345 


4156 


13682 


85432 


8793 


9823 


7532 


75671 


36457 


701 


6023 


9890 


86317 


21836 


9804 


1367 


6821 


24328 


17354 


1306 


8965 


9854 


98713 


63542 


717 


8632 


3821 


21345 


78163 


2103 


1034 


5843 


1286 


82645 


6397 


6312 


1936 


78654 


34685 


1096 


4593 


7136 


19876 


31768 


2130 


3687 


9876 


93643 


65314 


3107 


5006 


2863 


6356 


68231 


167 


7164 


123 


78397 


64037 


2109 


1763 


7436 


21602 


34685 


3678 


2139 


1567 


71346 


35962 


2176 


8236 


2563 


28653 


21363 


5432 


7860 


8432 


17648 


78636 


2137 


3613 


1345 


82351 


19854 


28639 


109 


8736 


21368 


80145 


1765 


1756 


8654 


78631 


87654 


371 


6386 


1263 


17639 


12345 


71031 


9890 


1345 


82360 


78654 


1463 


8243 


3093 


45671 


12345 


3168 



42. Find the sum of all the odd numbers under 100. 

43. Find the sum of all the even numbers under 100. 

44. Find the sum of all the numbers included in Ex. 
5, page 31. 

45. Find the sum of all in Ex. 6. 

Answers promiscuously arranged: 181217, 1300099, 
126362, 1325672, 136751, 143267, 52850, 2500, 2450, 
10100, 2510. 



ADDITION. 39 

46 to 49. Find the sum of all in Ex. 9, 10, 11 and 12.* 
50 to 53. Find the sum of all in Ex. 13, 14, 15 and 16. 
54 to 57. Find the sum of all in Ex. 17, 18, 19 and 20. 
58 to 60. Find the sum of all in Ex. 21, 22 and 23. 
61 to 64. Find the sum of all in Ex. 24, 25, 26 and 27. 

65. Add together all the numbers from 300 to 320, in- 
clusive; from 3120 to 3150, inclusive; from 160 to 200, 
inclusive; from 1950 to 2000, inclusive. 

Answers in direct order: 9615, 19228, 25576, 25512, 
19200, 211800. 

17. To add Federal Money, we place dollars under dol- 
lars, cents under cents and mills under mills, and proceed 
as before. 

66. What is the amount of the following sums of 
money? $32.74, $16.73, $13.09, $37.40, $16.74, $7.07. 

Am. $123.77. 

OPERATION. EXPLANATION. The sum of the first column being 27 
$32.74 cents, we write the 7 and add the 2 tens of cents to the 

16.73 tens' column, making 27 tens, or 2 hundreds and 7 tens. 
13.09 Writing the 7, we add the 2 hundreds to the next, which 
37.40 is the dollar column, and proceed as in the above. 

16.74 The second column of cents might be called the dime*' 
7.07 column. 

$123.77 Amount. 

67. Find the amount of the following: $1708.25, 
$2076.00, $709.07, $109.88, $999.87, $370.04, $695.83, 
$797.00, $87.00, $400.40, $198.08, $109.65, $364.08, 
$217.00, $364.09, $785.66, $699.08, $776.08. 

*The Teacher can work most of these by Arithmetical Progres- 
sion. As indicated by the numbers, these exercises may be broken 
into three or four parts, if considered too difficult. 



40 NELSON'S COMMON-SCHOOL ARITHMETIC. 

G8. $1670.03+81006.01+ $364.01+85432.99 

$2310.00+$1068.24+$26107. 18+82136.18 

$109.79+ $999.99+ $666.56+ $449.99 

$777.00+$7999.00+ $6666.00+86730.15 

69 70 71 72 

$987.67 $716.27 $187.20 $4519.27 
873.35 855.60 257.65 7864.20 

473.92 219.76 330.17 9510.33 

187.87 912.67 700.00 3578.84 
119.16 107.30 175.85 4875.60 
160.97 87.60 150.50 6115.90 
176.01 101.19 37.50 9885.10 
634.16 808.08 57.63 1105.75 

585.26 981.61 109.87 5760.87 
458.39 225.00 987.05 7901.57 

385.93 811.29 1285.58 7119.85 
589.38 300.92 2327.88 5006.29 
107.20 10.15 8900.17 9110.11 

70.99 106.30 209.18 4362.17 

18.18 547.67 101.01 3210.18 

1764.18 336.44 125.00 2133.64 

397.27 176.33 117.45 1364.57 
444.99 1275.84 361.45 2136.06 
222.66 666.57 217.33 1456.27 

799.88 1176.22 163.77 376.22 

73. Find the amount of the following sums of money: 
One hundred and twenty-five dollars and twenty-five 
cents; Sixty-eight dollars and forty-seven cents; Three 
hundred and ten dollars and eighty-seven cents j Six hun- 
dred dollars and seven cents; Four thousand eight hun- 
dred and fifty dollars and eighteen cents. 

Answers arranged promiscuously, including those to Ex. 
67 and 74: $11467.06, $64493.12, $45178062.31, $5954.84, 
$2624.91, $10422.81, $16802.24, $97392.79, $9457.42, 
$96394.79. 



ADDITION. 41 

74. Add together the following amounts: 

Eighteen thousand one hundred and forty-six dollars; 
Seven thousand one hundred and sixteen dollars and 
twenty-five cents; Sixty-four thousand one hundred dol- 
lars and four cents; Forty-five millions and one thousand 
dollars; Eighty-seven thousand seven hundred dollars and 
two cents. 

75. A merchant has 29 pieces of silk in 1 package, 35 
in another, 79 in a third. In the first there are 1497 
yards; in the second, 2173 yards; in the third, 4130 yards. 
How many pieces, how many yards? 

SOLUTION. 29 1497 

35 2173 

79 4130 



Whole number of pieces 143 Whole number of yds. 7800 

*76. A coal dealer sells 1254 bushels every day of the 
week, Sunday excepted; how many does he sell in all? 

To THE TEACHER. Columns of fifteen or twenty numbers may 
now be dictated to classes, the teacher observing to increase the 
speed of the scholars at every effort. The results may be called 
off as produced, and written by the teacher on the blackboard, or 
the learners may exchange slates for examination and correction. 

In this, as in all competitive exercises, the teacher should not 
wait until every member of the class has finished the work; but 
the tardy ones must not be overlooked, nevertheless. Means should 
be adopted to stimulate them to greater effort. They must be 
taught that they can not be allowed to fall behind without the 
risk of being returned to a lower class or grade. 

The teacher probably knows that to make boys or girls reckon 
rapidly he must lead; and to this end, it would be to the advantage 
of both teacher and pupil if such exercises as these were always 
impromptu. 

*In giving the answers, the learner should state whether it i3 
bushels, pounds, etc. The abbreviation Ibs. stands for pounds. 
4 



42 NELSON'S COMMON-SCHOOL ARITHMETIC. 

77. A merchant bought 9 bags of coffee, each bag 
weighing 215 Ibs., at an average cost of $21.20; what 
weight of coffee did he buy, and how much did he pay 
for it? 

78. A farmer has 118 sheep, 518 hogs, 210 pair of 
chickens, 5 plows, 6 wagons, 1 dozen hoes, 7 horses, 10 
spades and 12 pitchforks; how many animals has he, and 
how many agricultural implements? 

79. How many pupils in a school in which there are 5 
classes, the first containing 19, the second 28, the third 
32, the fourth 35 and the fifth 29 pupils ? 

80. Bought three boxes of oranges, in one of which 
there were 450, in another 469, in the last 510 oranges; 
how many did I buy? 

81. A man walked twenty-five miles on the 20th day 
of the month, twenty-three on the 22d, twenty-nine on the 
23d, thirty-three on the 24th day; how many miles did 
he walk altogether? 

82. How many days in the first nine calendar months 
of the year ? 

83. Sir Isaac Newton lived 85 years and was born in 
1642; in what year did he die? 

Answers: 143, 1429, 273, 110, 274, 1727, 7524, 1935, 
19080, 1063, 45. 

The Teacher should give numerous exercises besides thesd, and 
have his scholars work them on the blackboard before the class. 



HOME EXERCISES. 

1. Add 1, 2, 3, 4, 5, 6, 7, 8 and 9 to 2, and take them from 
the result again, writing them out as below : 
2 and 1 are 3; 2 from 3 leaves 1. 
2 and 2 are 4; 2 from 4 leaves 2. 
2 'and 3 are 5; 2 from 5 leaves 3. 



SUBTRACTION. 43 

2. Add and subtract (take from) 3 in the same way. 
3 Add and subtract 4 in the same way. 

4. Add and subtract 5 in the same way. 

5. Add and subtract 6 in the same way. 

6. Add and subtract 7 in the same way. 

7. Add and subtract 8 in the same way. 

8. Add and subtract 9 in the sam^ way. 



IV. SUBTRACTION. 

18. The process of finding the difference between two 
numbers is called Subtraction. 

19. This difference is called the remainder or excess. 

20. The sign is , and is called minus. When placed 
between two numbers, it shows that the one on the right 
is to be taken from the one on the left : 7 5=2, reads, 
seven minus Jive equals two. 

1. To find the difference between 375 and 263. 
SOLUTION. 1. Writing the small number under the 375 

large one, units and tens of the one under those of the 263 

other, we proceed to subtract 3 from 5, which leaves 2; 

this we write under the 3. 112 

2. 6 from 7 leaves 1 ; write it under the 6. 

3. 2 from 3 leaves 1. The remainder is 112. 

2. From 186436 take 165213 An*. 21223 

3. From 786900 take 654300 132600 

EXERCISES FOR THE BLACKBOARD. 

When subtracting one figure from another, the learner should be 
taught to see the result, rather than to reckon it or talk over the 
process. This can be done by such exercises as the following : 

Taking a row of figures as 4903781236542, point to 9, requiring 
the class to give the difference between it and 4 ; th,en to 0, requir- 
ing the difference between 10 and 9, etc. 



44 NELSON'S COMMON-SCHOOL ARITHMETIC. 

4. From 49368282 take 15012 Am. 49353270 

5. 247896136785 8. 66145397 134286 

6. 716035 15012 9. 151764824164271 

7. 371150 70000 10. 37898643 276321 

Total, 1113284 Total, 114645644 

11. 1317^3^6^87^45 13145363435 

12. 984960997610899771900986010098 

13. 19899799994896 7445199821886 

Total, 238677822898021. 

When some of the figures of the smaller number are 
greater than those above them, we add ten to both figures. 

14. From 342 take 267. 

To subtract the 67 from the 42 above it, we add 10 to 10 

3J. V 

both numbers, as indicated by the small figures; but in- 
stead of adding to both numbers in the same place, we 267 
add 10 to the unit 2 of the upper number, and 1 (ten )to 
the 6 tens of the lower number. ' ^ 

PROCESS. 1. We can not take 7 from the 2; add 10 which makes 
12; 7 from 12 leaves 5, which write. 

2. Adding 1 (ten) to the 6 we have 7, which taken from 14, after 
adding another ten (or one hundred) leaves 7. 

3. Adding 1 (hundred) to the 2 we have 3, which taken from 3 
leaves nothing. The remainder is 75. 

21. PROOF. By adding the remainder to the smaller 
number, we should get a sum equal to the larger. 

In the above example the remainder was 75, the 
smaller number 267, which added together equal the *?& 

larger, 342. 

*15. 1603845732164000000 

98123456789798768 



1505722275374201232 Rem. 



*The Teacher will require the class to read off both question and 
answer. 



SUBTRACTION 45 

16. From 31642789600 take 1278899765. 

Rem. 30363889835. 

17. From 16782466987 take 123469978. 

Rem,. 16658997009. 

18. 216836732178491637,642178 

19. 2216821783680011000999999 

20. 100000000000000 1 

21. 3681068213682 11194680 

Total, 347044269962674. 

22. From $1670 take $389.27. 

OPERATION. $1670.00 

389.27 



$1280.73 Am. 
When writing the following questions, be particular to 

To THE TEACHER. Mental exercises in adding will be found a 
good means of cultivating the retentive faculty for business pur- 
poses. Such exercises should not consist of single digits nor of 
fractions with terms made up of single digits, but of numbers such 
as the following: 

27 and 64 are how much? To add these numbers the tens should 
be taken first: 27 and 60^87 and 4=91. 

$3.27 and $1.25 are how much? . 327 and 120=447 and 5=452. 

COMPETITIVE EXERCISES 

In addition might come in here, and be introduced at intervals 
throughout the course. A problem being written on the black- 
board, or dictated to the class, scholars should be required to hand 
in their slates in the order in which they obtain the results, when, 
the teacher would number them, "1, 2, 3, wrong, 4, 5," etc., calling 
the name of the competitor in each case and returning the slatei 
Such exercises should be conducted by the teacher with celerity, so 
that at a single glance he can tell whether the learner has the cor- 
rect result. Fifteen minutes will be found sufficient time to devote 
to such exercises at once. 



46 NELSON'S COMMON-SCHOOL ARITHMETIC. 

arrange dollars under dollars and cents under cents. 
When there are no cents, write two ciphers in their place. 

23. $10067.89 $2141.98 27. $60000.00 $4670.87 

24. $15070.14 6160.47 28. $23678.45 4101.00 

25. $1001000. 1.86 29. $1006812.00 3178.59 

26. $6743147. .97 30. $678997.00 210.99 
Total, $7760979.75. ' Total, $1757326.00. 

31. $710356.87 $14683.29 35. $68750.37 $1416.44 

32. $370968. 17987.77 36. $71000.90 87.50 

33. $478979. 14780.99 37. $100000. .87 

34. $100000. 374.66 38. $61987.15 .99 
Total dif., $1612477.16. Total dif., $300232.62. 

^39. What is the sum and difference, when added to- 
gether, of $36748..94 and $10968.75. 

40. Borrowed of A, at different times, $146.87, $6740.18, 
$310.75, and have paid him $10.00, $450.18 and $61.14; 
how much do I owe? 

41. Out of 5 hogsheads of sugar containing 5761 Ibs., 
I sold 3, containing 1114 Ibs., 1311 Ibs. and 1001 Ibs.; 
how much was left? 

42. After selling 1347 Ibs. of sugar from 3 hogsheads, 
each containing 1000 Ibs., how much was left? 

EXERCISES IN TAKING THE COMPLEMENT OB "MAKING 
CHANGE." 

To THE TEACHER. Taking $1 as the complete number, require 
the complement of 25 c, 27 c, 30 c, 35 c, etc. It 'will be found that 
the complement of the teens is in the 80s, of the 20s in the 70s, of 
the 30s in the 60s, of the 40s in the 50s, of the 60s in the 30s, of 
the 70s in "the 20s, etc. "54?" Aris. 46. 35? Ans. 65. 

Taking $2, $3 or $5, the exercise may be practiced in the same 
way. 

*Give the denomination of the answer, whether it be dollars, 
pounds, etc. 



SUBTRACTION. 47 

43. A merchant, owns goods to the amount of $3147, 
and lands to the amount of $2107, and is indebted $1400 : 
to A $200, to B $340 and to C $860; what is the amount 
of his net capital ? 

44. A merchant sells goods for another to the amount 
of $4374.23, and is to receive $43.75 for his trouble, be- 
sides the expenses of freight, etc., which was $125.15 ; 
how much should he return to his principal? 

45. What is the difference between 1856 and 1798? 
When was the individual born who died in 1857 at the 
age of 45 years? When will the work be completed 
which was commenced in 1855 and was to take eighteen 
years ? 

Answers to the last seven: 420533, 58, 1812, 2454, 
1873, 1653, 7349788, 2335, 667648, 410533 .* 

*Give the denomination of the answer, whether it be dollars, 
pounds, etc. 

HOME EXERCISES/}- 



1. 30+25=? 


14. 78+65=? 


27. 94+27=? 


2. 27+40=? 


15. 59+63=? 


28. 79+86=? 


3. 26+30=? 


16. 72+48=? 


29. 56+65=? 


4. 43+27= ? 


17. 37+75=? 


30. 33+63=? 


6. 29+12=? 


18. 59+76=? 


31. 98+63=? 


6. 54+16=? 


19. 85+64=? 


32. 27+79=? 


7. 33+17=? 


20. 78+77=? 


33. 56+65=? 


8. 55+75=? 


21. 99+37=? 


34. 73+64=? 


9. 47+43=? 


22. 55+66=? 


35. 55+68=? 


10. 82+12=? 


23. 37+74=? 


36. 97+63=? 


11. 95+15=? 


24. 65+37=? 


37. 64+67=? 


12. 34+45=? 


25. 59+63=? 


38. 73+67=? 


13. 29+64=? 


26. 37+65=? 


39. 45+63=? 



t To be written on the slate at homeland the operations recited in 
Bchool. 



48 NELSON'S COMMON-SCHOOL ARITHMETIC. 

40. 27-f-33=? 60. 59+16=? 80. 97+63=? 

41. 95+64=? 61. 27+17=? 81. 84+37=? 

42. 84+33=? 62. 86+59=? 82. 29+18=? 

43. 27+69=? 63. 75+38=? 83. 63+97=? 

44. 55+69=? 64. 64+55=? 84. 25+88=? 

45. 78+73=? 65. 29+33=? 85. 36+54=? 

46. 65+63=? 66. 36+93=? 86. 97+79=? 

47. 73+39=? 67. 74+67=? 87. 88+69=? 

48. 55+97=? 68. 36+56=? 88. 53+76=? 

49. 77+67=? 69. 75+27=? 89. 95+84=? 

60. 59+63=? 70. 33+13=? 90. 67+96=? 

61. 78_|_64=? 71. 29+19=? 91. 44+39=? 

62. 59+33=? 72. 98+29=? 92. 29+88=? 

63. 98+97=? 73. 29+35=? 93. 74+86=? 
54. 65+36=? 74. 18+62=? 94. 49+57=? 
65.21+17=? 75.79+84=? 95.64+88=? 

66. 76+93=? 76. 56+33=? 96. 98+77=? 

67. 54+64=? 77. 49+54=? 97. 55+76=? 

68. 79+98=? 78. 59+39=? 98. 74+84=? 

69. 63+64=? 79. 28+97=? 99. 85+99--^? 

Write the multiplication table as follows: 

1. 2 times 1 or once 2 is 2. 
2 times 2 are 4. 

2 times 3 or 3 times 2 are 6. 
2 times 4 or 4 times 2 are 8. 
Continue this to 12. 

2. Write the 3 times table to 12 in the same way. 

3. Write the 4 times table to 12 in the same way. 

4. Write the 5 times table to 12 in the same way. 
6. Write the 6 times table to 12 in the same way. 

6. Write the 7 times table to 12 in the same way. 

7. Write the 8 times table to 12 in the same way. 

8. Write the 9 times table to 12 in the same way. 

9. Write the 10 times table to 12 in the same way. 

10. Write the 11 times table to 12 in the same way. 

11. Write the 12 times table to 12 in the same way. 



MULTIPLICATION. 



IV. MULTIPLICATION 

22. Multiplication is a short method of adding, when 
the same number has to be repeated any number of times. 
X is the sign. 3X^18, reads, three times six equals 
eighteen. 

1. To find the sum of 123+123+123, by additions we 
would enter the three amounts as before, and add for the 
result. 

In multiplication, we multiply each figure of the num- 123 

ber to be increased by the number which indicates 3 

liow often the repetition is to be made, thus: . 

3 times 3 are 9; put 9 in the unit's place. 369 

3 times 2 are 6; put 6 in the ten's place. 

3 times 1 are 3, which put in the hundred's place. The result is 
869, as it would have been by addition. 

TERMS. 

23. The number to be multiplied is called the multipli- 
cand, the number by which it is multiplied, the multiplier, 
and the number produced by multiplying, the product. 
The multiplicand and multiplier are also called factors. 

T, (123 Multiplicand. 

FACTORS | jj 



369 Product. 
2. To find the product of 1496 by 7. 

Here we say 7 times 6 are 42; write 2 under the 7. 1496 

Then 7 times 9 are 63, and the 4 we carried make 67; 7 

write 7 and carry 6. 7 times 4 are 28 and G are 34; - 

write 4 and carry 3. 7 times 1 are 7 and 3 are 10. 10472 

Ans. 10472 



50 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



3. 2146X2= 4292 

4. 3178X3= 9534 

5. 4167X4=16668 

9. 5189X5=? 

10. 7864X6=? 

11. 2875X7=? 

Sum, 93254. 



6. 21007X5=? 

7. 31497X<5=? 

8. 17843X7=? 

Sum, 418918. 

12. 41679X 8=? 

13. 98765 X 9=? 

14. 73149X12=? 

Sum, 2100105. 



Observe to point off the cents in the products of the 
following : 



15. $21.37X7=? 

16. $117.49=8=? 

17. $317.00x0? 
Amount, $3942.51. 

21. $678.39X11=-? 

22. $467.28X12=? 

23. $999.99X 9=? 
Amount, $22069.56. 

27. $47.531 X 9=? 

28. $716.145X11=?' 

29. $9871.321x12=? 
Amount, $126761.226. 

33. $9057.179X12=? 

34. $7898.796X 9=? 

35. $5970.463X11=? 
Amount, $245450.405. 

39. $7161.213x11? 

40. $1409.796X12=? 

41. $9393.678 X 9=? 

42. $4131.196X10=? 

Amount. $221545.957. 



18. $10.73X9? 

19. $117.07X6=? 

20. $307.49x7=? 
Amount, $2951.42. 

24. $671.49X10=? 

25. $857.37X11? 

26. $1096.49X12=? 

Amount, $29303.85. 

30. $1670.053x7=? 

31. $2199.989X9=? 

32. $7186.739x8=? 

Amount, 88984.184. 

36. $793.179X9=? 

37. $987.970X8=? 

38. $213.219X7=? 
Amount, $165349.04. 

43. $3714.291 X^? 

44. $7965.379X8=? 

45. $3768.219x7=? 

46. $ 419.367X6=? 
Amount, $126045.386. 



MULTIPLICATION. 51 

47. 2785X357. 

We have here three multipliers seven, fifty and three 
hundred. 

2785X7= 19495 19495 

2785X5 tens= 13925 tens, or 139250 

' 2785x3 hundreds= 8355 hundreds, or 835500 



Total products, 994245 

This operation is contracted by arranging the figures 2785 
as in the margin, and writing the first figure of the 357 

products of the units in the unit's place and the others 
to the left, of it; the first figure of the product of the 



tens in the ten's place, or under its own multiplier, 5; oore 
and the first figure of the product of the hundreds in the 
hundred's place. 994245 

Either factor may be used as a multiplier in the fol- 
lowing exercises? 

48. 3170X 178=? 51. 2896x6789=? 

49. 6184X1794=? 52. 7109x9998=? 

50. 3867X3784=? 53. 2345x3979=? 
Total, 26291084. Total, 100067481. 

To THE TEACHER. Blackboard exercises in concert may be given 
in the following manner: 

Writing a line of figures, 379463875426, the teacher would lead 
by multiplying the second by the first, the third by the second, 
the fourth by the third, etc., without speaking the process. Pointing 
to 7, he would say 21; to 9, 63; to 4, 36, etc. 

To instruct in "carrying," the same line may be used by point- 
ing to the third figure and performing the following operation 
mentally: 3X"-f 9=30. Pointing to 4, he would say 67; to 6, 42; 
to 3, 27. 

To produce rapidity of thought and action, exercises of this kind 
ought to be frequent, and the teacher should lead, taking care that 
the whole class follows. 

Such exercises as these may be profitably continued throughout 
the entire course of study. 



52 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



54. 6789X2164=? 

55. 1578X 753=? 

56. 9409X6781=? 

57. 2783X4679=? 

Total, 92703716. 

62. 420001000 
109608 



58. 8976X7659=? 

59. 3968X6483? 

60. 7689X2197=? 

61. 6784X7898=?. 

Total, 1649444493. 

109608 
420001000 



3360008000 
2520006 
3780009 
420001 



109608 
219216 
438432 

Product, 46035469608000 



Product, 46035469608000 

The multiplier of the ten's place in the first operation 
being 0, we passed it, and multiplied by the 6 hundreds. 
In the second operation we passed the ten's, hundred's 
and thousand's places for the same reason.* 



63. 12346X30010=? 

. 64. 7684X10900=? 

65. 6787 X 3009=? 

Total, 474681143. 

69. 2000X 7010=? 

70. 3160X10096=? 

71. 2178X90909=? 

Total, 243923162. 



66. 4967X 6007=? 
67.. 5896X900707=? 
68. 7649X 66080=? 

Total, 5845851161. 

72. 1009x90910=? 

73. 21678X21006=? 

74. 31784X 7009=? 

Total, 769870314. 



24. To multiply ~by 10, 100, 1000, etc., we have simply 
to annex as many ciphers to the multiplicand as there 
are in the multiplier. 



*If the learner will simply observe to write the first figure of each 
product under its own multiplier, he will have no difficulty in mul- 
tiplying where there are ciphers. For instance, the first figure of 
the product by 2, in the second example, is immediately under the 2. 



MULTIPLICATION. 53 

35X10=350. 
Proof 35 

10 Ten times 5 are 50, and 10 times 3 are 30 and 5 

are 35, making 350. 

350 

75. 1G5X 10= 1650 78. 413X 10=? 

76. 165X 100= 16500 79. 1716X 100=? 

77. 165x1000=165000 80. 9417x1000=? 

Total, 9592730. 

81. 374X 100=? 84. 9361 X 10=? 

82. 268X 1000=? 85. 7342X 100=? 

83. 189X10000=? 86. 8654x1000=? 

Total, 2195400. Total, 9481810. 

25. To multiply dollars, cents and mills, we remove 
the decimal point to the right. 

87. What is the product of $279.373 by 10? 

Am. $2793.73. 

EXPLANATION. By multiplying the mills by 10 we make them 
cents, by multiplying the cents by 10 we make them dimes, and by 
multiplying the dimes by 10 we make them dollars. 

88. $145.373 XlOO=$14537.3, or 14537 dollars and 3 
dimes or 30 cents. 

89. $356.14,5 X 10=? 92. $317.98,7x 100=? 

90. $178.91,3X100=? 93. $679.97,6X 10=? 

91. $463.97,8X100=? 94. $7193.44,5x1000=? 

Amount, $67850.55. Amount, $7232043.46. 

95. $713.71,4X 100=? 99. $131.71,2x1000=? 

96. $165.79,3X1000=? 100. $724.26,8 X 100=? 

97. $786.47,5X 10=? 101. $413.16,4X 10=? 

98. $130.14 X 100=? 102. $236.21 X 100=? 
Amount, $258043.15. Amount, $231891.44. 

To THE TEACHER. Mental exercises on this subject should suc- 
ceed these written ones. 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



HOME EXERCISES. 

I. Commencing at 13, write the 2 times table to 19 in 
this way : 

2 times 13 are 26; 13 times 2 are 26. 
2 times 14 are 28 ; 14 times 2 are 28. 

2. Write the 3 times table in the same way. 

3. Write the 4 times table in the same way. 

4. Write the 5 times table in the same way. 

5. Write the 6 times table in the same way. 

6. Write the 7 times table in the same way. 

7. Write the 8 times table in the same way. 

8. Write the 9 times table in the same way. 

9. Write the 13 times table to 9 as follows: 

13 times 2 are 26; 2 times 13 are 26. 

13 times 3 are 39; 3 times 13 are 39. 

10. Write the 14 times table in the same way. 

II. Write the 15 times table in the same way. 

12. Write the 16 times table in the same way. 

13. Write the 17 times table in the same way. 

14. Write the 18 times table in the same way. 

15. Write the 19 times table in the same way. 

1. Write the 2 times table to 19 as follows: 

2 times 3 are 6; 2 times 13 are 26. 
2 times 4 are 8; 2 times 14 are 28. 
2 times 5 are 10; 2 times 15 are 30. 

2. Write the 3 times table in the same way. 

3. Write the 4 times table in the same way. 

4. Write the 5 times table in the same way. 

5. Write the 6 times table in the same way. 

6. Write the 7 times table in the same way. 

7. Write the 8 times table in the same way. 

8. Write the 9 times table in the same way. 



MULTIPLICATION. 55 

EXERCISES IN MULTIPLYING THE " TEENS."* 

100. 2357X13 and 14? 104. 7890x10 and 18? 

101. 5398X15 and 16? 105. 2164x17 and 16? 

102. 6532X17 and 18? 10(1. 3165x16 and 15? 

103. 7654X17 and 19? 107. 2137x15 and 14? 
Total products, 735141. Total products, 523430. 

108. $45.67X16 and 15? 112. $76.54 x!6 and 17? 

109. 14.59X15 and 14? 113. 57.352x19 and 15? 

110. 23.08X13 and 14? 114. 67.185x18 and 17? 

111. 21.87X17 and 19? 115. 45.375x16 and 18? 
Total products, 3249.36. Total products, 8370.013. 

116. 137.67 Xl7andl8? 120. 43.165X10 and 16? 

117. 216.031X15 and 18? 121. 933 Xl8andl5? 

118. 131.75 Xl6andl6? 122. 61.751x16 and 17? 

119. 231.35 Xl7andl9? 123. 311.155x14 and 13? 
Total prod., 84595.96. Total prod., 42738.743. 

f PRINCIPLES OF MULTIPLICATION. 

26. When two numbers are to be multiplied together, 
we use for the multiplier that which will produce least 
figures in the operation. This will be accomplished by 
selecting the smaller number, except where there are many 
ciphers, as in Ex. 62. 

27. If a number of articles and the price of one article 
be multiplied together, the product will be the price of 
all at the same rate. 

3 yards of muslin at 20 cents. 

20X'3=60 cents. 

If the price of one be in cents, the price of all wilf be 
in cents. If in dollars, the price of all will be in dollars. 

* Every boy designed for business pursuits ought to commit to 
memory the multiplication table up to 19 times, inclusive. 



56 NELSON'S COMMON-SCHOOL ARITHMETIC. 

28. The number of articles contained in any box, bale, 
package, etc., multiplied by the number of boxes, bales, 
etc., each containing a like number, will give the number 
of articles in all. 

In a box there are 30 articles; how many in 20 such 
boxes? 

30X20=600 articles. 

29. A number multiplied by itself is squared, or raised 
to the second power, and the second power multiplied by 
the same number is cubed, or raised to the third power. 
The sign is a small figure on the right of the number, 
thus, 5*, which indicates that 5 is to be raised to the 
fourth power, and is equal to 5x5X^X5, or 625. 

30. Feet multiplied by feet and yards multiplied by 
yards produce square feet and square yards. 

12 feetXl2 feet=144 square feet. 

31. Any number of feet multiplied by the number of 
inches in one foot will give the number of inches in all 
the feet. Pounds multiplied by the number of ounces in 
one pound will give the number of ounces in all the 
pounds, and so with numbers of any other denomination. 

How many inches in 37 feet? 

37X12=444 inches. 

1. What is the price of 37 bushels of corn at 37 cents 
per bushel? 

2. What should I pay for 357- yards of broadcloth at 
$2.?5 per yard? 

3. Find the cost of 325 acres of land at $57 per acre. 

4. In 320 bales of cotton there are 460 Ibs. each ; how 
many in all? 

5. In 557 pieces of muslin there are 35 yards each; 
how many in all? 



MULTIPLICATION. 57 

6. A ship laden with flour has 7950 barrels on board, 
and in each barrel there are 196 Ibs.; how many pounds 
in all? 

7. In a bushel of dried apples there are 25 pounds; 
how many are there in 37 bushels? 

8. A barrel of flour weighs 196 pounds; what is the 
weight of 325 barrels? 

9. What will be the weight of 134 bushels of wheat 
when 60 Ibs. are allowed to the bushel? 

10. Find the cost of 379 boxes of cheese, each of which 
weighs 22 pounds, at 25 cents a pound. 

11. A box of buttons contains a gross; how many but- 
tons are there in 59 boxes? 

12. A merchant sold 135 barrels of flour at $6.75 apiece, 
and with part of the money bought 369 bushels of coal 
at 25 cents; how much money had he left? 

13. In 236 yards how many inches? 

14. How many quarts are there in 27 bushels? 

15. At 23 cents a quart, how much money can be real- 
ized on 18 bushels of strawberries, allowing one quart for 
loss in measuring? 

16. A huckster bought two barrels of apples, each con- 
taining 3 bushels, at $6 a bushel, and sold them at 87 
cents a half peck ; allowing half a peck for loss in meas- 
uring, did he gain or lose? and how much? 

17. There are 12 inches in a foot and 3 feet in a yard; 
how many inches are there in 357 yards? 

18. How many quarts of wine are there in 6 barrels, 
each of which contains 42 galls.? how many in 35 bar- 
rels? how many in 163? 

Answers without their denominations: 13225, 864, 819, 
8496, 208450, 8496, 8040, 925, 98175, 21397, 18525, 
147200, 63700, 19495, 1558200, 489, 1369, 12852, 34272. 



58 NELSON'S COMMON-SCHOOL ARITHMETIC. 

19. Find the cost of 117 bushels of apples at 35 cents 
a peck. 

20. Find the cost of 237 bushels of potatoes at 42 cents 
a peck. 

21. Bought 46 horses at 125 apiece, and sold them 
for 85900; did I lose or gain, and how much? 

22. At 5 cents apiece, what will 22 gross of eggs cost? 

23. In 15 acres how many square yards? 

24. Find the cost of 35 barrels of molasses at 33 cents 
a gallon, each barrel containing, on an average, 41 galls. 

25. In 257 cords of wood how many solid feet? 

26. How many bushels of coal in 17 wagons, each car- 
rying 50? and what will be the cost at $17 a load? 

27. How many yards are in a box of silk containing 35 
pieces, each piece measuring 52 yards? 

28. In a case of muslin there are 45 pieces, each 31 
yards; what will be the cost of it at 55 cents a yard? 

29. A bushel of hemp-seed weighs 44 pounds; what 
will be the weight of 137 bushels? 

30. How many leaves are in 67 reams of paper? 

31. How many half-pence are in 527 pounds sterling? 
Answers promiscuously arranged: 72600, 16380, 5175, 

150, 15840, 39816, 47355, 64320, 76725, 14450, 1820, 
32896, 6028, 252960. 



HOME EXERCISES. 



1. 3+ 4 2=1 
2. 5 3+ 7=? 
3. 9>< 4_ 3 = ? 
4. 7>< 6+ 8=? 
6. 6+ 9X 6r=? 
6. 9 8 1=1 


7. 7+ 3X 6=? 
8. 9-f- 8+ 3=? 
9. 9>< 7_ 4^? 
10. 5+ 5 3=? 
11. 8X 4+ 2=1 
12. 9 3 2=1 


13. 16+12 5=,? 
14. 23-}-20 1=? 
15. 1916+ 1=1 
16. 2712-1- 8r=? 
17. 29+ 2 7=1 
18. 40+54 3? 



MULTIPLICATION. 



19. 35 15X 6=? 

20. 64 -fSO 2=? 

21. 971?,+ 7=? 

22. 55+43 8=? 

23. 63+65+ 2=1 

24. 19+ 6 5=? 

25. 29+ 6 5=? 

26. 35+25+15=? 

27. 18+2210=? 

28. 54+16+10=? 

29. 93+1720=? 

30. 44+16+32 = ? 

31. 12+18+33=? 

32. 27+33+54=? 

33. 44+1615=? 
84. 1912+28=? 
36. 55+44+21=? 

36. 23+60+47=? 

37. 64+15+36=? 

38. 24+46+33=? 

39. 56+44+71=? 

40. 19+33+21=? 

41. 77+44+23=? 

42. 45+31+55=? 

43. 37+13+34=? 

44. 66+1412=? 

45. 44+76+33=? 

46. 92+1810=? 

47. 73+10+67=? 

48. 21+39+22=? 

49. 17+63+14=? 



50. 29+33+19=? 

51. 3X15+15=? 

52. 7+16X 3=? 

53. 57 17X "^? 

54. 19+11X 6=? 

55. 45+15X 9=? 

56. 84+26X12=? 

57. 21+18+15=? 

58. 45+1611=? 

59. 73+1316=? 

60. 55+45+35=? 

61. 79+60+30=? 

62. 24+71+ 5=? 

63. 39+1110=? 

64. 9919+12=? 

65. 77+7914=? 

66. 85+9611=? 

67. 77+6320=? 

68. 45+6522=? 

69. 33+7320=? 

70. 97+93+50=? 

71. 79+91+93=? 

72. 56+64+49=? 

73. 73+65+37=? 

74. 93+65+35=? 

75. 97+63+47=? 

76. 45X 3+45=? 

77. 35X 4+30=? 

78. 29X 614=? 

79. 99X 327=? 

80. 84 14X 7=? 



81. 75X 3 6=? 

82. 36X 414=? 

83. 56X 3+50=? 

84. 75 16X 6=? 

85. 35X1220=? 

86. 42X10+22=? 

87. 57X1214=? 

88. 4515X30=? 

89. 64X 520=? 

90. 18X 628=? 

91. 36X12+72=? 

92. 42X12+12=? 

93. 35X20+12=? 

94. 64X3214=? 

95. 33X2112=? 

96. 14X1213=? 

97. 12+33 6=? 

98. 75X16 4=? 

99. 39X12+ 4=? 

100. 5616+ 6=? 

101. 35X15X 6=? 

102. 3522X12=? 

103. 64+2210=? 

104. 75X9737=? 

105. 64X14 4=? 

106. 3616X96=? 

107. 5523X20=? 

108. 4919X84=? 

109. 64X4515=? 

110. 976717=? 

111. 37+16 6=? 



1. Write the 2 times table to 9 as follows: 

2 times 2 are 4 ; 2 is contained in 4, 2 times. 
2 times 3 are 6; 2 is contained in 6, 3 times. 
2 times 4 are 8; 2 is contained in 8, 4 times. 

2. Write 3 times table to 9 as above. 

3. Write 4 times table to 9 in the same way. 

4. Write 5 times table to 9 in the same way. 
6. Write 6 times table to 9 in tlie same way. 



60 NELSON'S COMMON-SCHOOL APITHMETiC. 

6. Write 7 times table to 9 in the same way. 

7. Write 8 times table to 9 in the same way. 

8. Write 9 times table to 9 in the same way. 

1. Write the division table, as below, to 2 in 19: 

2 in 2, 1 time. 

2 in 3, 1 time and 1 left. 

2 in 4, 2 times. 

2 in 5, 2 times and 1 left. 

2. Find how often 3 is contained in numbers from 3 to 29. 

3. Find how often 4 is contained in numbers from 4 to 39. 

4. Find how often 5 is contained in numbers from 5 to 49. 

5. Find how often 6 is contained in numbers from 6 to 59. 

6. Find how often 7 is contained in numbers from 7 to 69. 

7. Find how often 8 is contained in numbers from 8 to 79. 

8. Find how often 9 is contained in numbers from 9 to 89. 



V. DIVISION. 

32. Division is the method of calculation used to sep- 
arate numbers into equal parts. 

33. Division may be short or long. It is short when 
the process of finding the product and remainder is per- 
formed mentally, and long when the process is written. 

34. The sign is -f-, which placed between two numbers 
indicates that the one on the left is to be divided by the 
one on the right. 

6-^3, reads, six divided by three. 

Division is also indicated by a curved line between the 
numbers, thus: 3)6; and by a straight line, with one 
number above and the other below, as |, called the frac- 
tional form. The period is also used to indicate division. 
.5 shows that 5 is divided by 10. 

35. TERMS. 
The number divided or to be divided is the dividend. 



DIVISION. 61 

The number by which the division is performed or to 
be performed is the divisor. 

The number which shows how many times the divisor 
is contained in the dividend is the quotient. 

The number left after dividing, the remainder. 

Divisor, 3)16784 Dividend. 



Quotient, 5594 2 Remainder. 

SHORT DIVISION. 

1. To divide 738 by 3. 

3)738 EXPLANATION. 1. Commencing at the left, we find 

- how often 3 is contained in 7 hundred, which is 2 hun- 
246 dred times, with a remainder of 1 hundred. The 2 

we write in the hundreds' place. 

2. This 1 hundred, with the 38, gives a remainder of 138 to 
be divided. To divide this, we consider the 13 as tens. In 13 
tens 3 is contained 4 times, and 1 ten left as a remainder ; so we 
write the 4 in the tens' place. 

3. This 1 ten, with the remaining 8, gives 18 to be divided. 
In 18, 3 is contained 6 times, which, being written under the 

8, gives the result, 246. 

Until he becomes familiar with the process, the learner 
may write the remainders in small figures, as in the fol- 
lowing example. 

2. 8)1 3 5 2 7 3 6 4 8 3 3 7 5 6 1 *7 



163460470 21 

EXPLANATION. 1. Commencing at the left we find how many 
times 8 is contained in 13. The answer being 1 time, with a re- 
mainder of 5, we write the 1 under the 13 and the 5 before the 0. 

2. This 5 taken with the makes 50, in which 8 is contained 6 
times, with a remainder of 2. 

3. This 2, with the 7, makes 27, in which 8 is contained 3 times, 
with a remainder of 3. 

4. When 3 in the dividend is reached, it is found that 8 is not 



62 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



contained in it, so a cipher is placed under it and 3 considered a 



remainder.* 

Divide the following: 

3. 134615379-4-2 

4. 21637298452-5-3 

5. 59368217755--4 

6. 1416823687949-f-5 

Rem 

7. 13645217--6 5 

8. 23176841-:-7 2 

9. 47896739--8 3 
10. 89765432-J-9 8 



Quotients. Rem. 

Am. 673076891 

Am. 72124328171 

Ans. 148420544383 

Am. 2833647375894 

Rem. 

11. 36174573110 1 

12. 21376495211 5 

13. 17896152112 1 

14. 34567890012 



Total quotients, 21546207 and 99327785. 

15. 80923751763=? 19. 33376231825 7=? 



16. 7316305416- 

17. 351923649755=? 

18. 13138539444 



23. 6542050675611=? 

24. 9645148784412=? 

25. 31463215726 7=? 

26. 21347096543 8=? 
Total, 2114807495312. 



-4=? 20. 72143250072 8=? 
21. 30608807751 9=? 
-6=? 22. 8172695439010=? 



Total quotients, 13754764315 and 25359613454. 



27. 57S312908-f-3=? 

28. 483796459-^-4=? 

29. 761147355-^-5=? 

30. 123450678-4-6= ? 
Total, 4865246673. 



31 to 34. Divide 874109630175 by 2, 3, 4 and 5. 

Sum, 11217740253904. 

* PROOF OF DIVISION. Division may be proved by multiplying 
the quotient by the divisor and adding the remainder. Take the 
last example: 

1634604702-1 






12076837617 



DIVISION 63 

35 to 38. Divide the same number by 6, 7, 8 and 9. 

Sum, 47694473868421. 
39 to 41. Divide the same number by 10, 11 and 12. 

Sum, 2397179440329. 

42. 710084973-r- 7=? 46. 31463574213--7=? 

43. 394789006-f- 8=? 47. 91678543210-=-8=? 

44. 361007839-5- 9=? 48. 76303074368--9=? 

45. 909738697--10=? 49. 21356703648-f-6=? 

Sum, 28187518617. Sum, 279921841995. 

36. The quotient of a number divided by 2 is the 

(one-half) of it ; divided by 3 it is the -| (one-third) ; by 

4, the \ (one-fourth); hence, to find the , \ or \ of a 

number, we have simply to divide by 2, 3 or 4.* 

Let it be requited to find the cost of 2| yards of cloth 
at $3 a yard. 

$3 =cost of 1 yard. 
2 

6 cost of 2 yards. 
\^=cost of \ yard. 

$7=cost of 2^ yards. 
Here, to multiply by 2 j, the 3 had to be divided by 2. 

50. -J of 3716=1238g f 54 - 7 of 34161143764= - 

61. -I of 1367= - 1 55. of 37897181237= - | 

62. ^ of 7854= - 1 56. J of 16872352168= - | 
53. J of 879= - 1 57. T V of 34564185432= - / 3 



*The learner will be particular to observe that finding the J or J 
of a number is not dividing by one-half or one-third, but simply 
finding one part of something divided into two or three parts which 
is multiplying by one-half or one-third. 

"j The remainder in this example being 2, we write it as f, which 
indicates that 2 is divided by 3, or that it is 2 parts of something 
divided into 3 parts. The remainders only of the questions which 
follow will be given. 



64 NELSON'S COMMON-SCHOOL ARITHMETIC. 

58. $14567.85X9| is how much? 
1456785 cents. 



13111065 
485595 

13596660 or $135966.60 

The decimal point was removed before dividing and replaced 
after the operation was performed. 

59. J- of $21372 =? 63. of $67849132.87=? 

60. of $13744 =? 64. \ of $16493178.00=? 

61. i of $73176.35=? 65. \ of $23610934.10=? 

62. \ of $14537.07=? 66. T V of $12310985.47=? 

Total, $25092.50. Total, $14579927.51. 

67. $345.78 X 37 ^=how much? 
34578 
37A 



17289 = \ of multiplicand. 
242046 = 7 times do. 
103734 =30 times do. 



1296675 =37-| 
or $12966.75. 

68 and 69. $146.82x8-*=? $1713.14x6{ = ? 

Sum, $11930.62,1. 
70, 71 and 72. $4563.28X451, 16J and 18. 

Sum of products, $365062.39,91+ 1. 
73, 74 and 75. $21763X14}, 15, 29|. 

Total, $1274430.916 c j. 
76, 77 and 78. $7649.14X76J, 96J, 86-j. 

Total, $1977090.20j + . 

79 to 81. $3146X2^, 6^, 12^. Total, $67639.00. 

82 to 84. $15G7X^, 5," llU. $39175.00. 

85 to 87. $7864X6|, 71, 37^. $400670.80. 



DIVISION. 



65 



88 to 90. $71684.25x8|, 7|, 5'. 
91 to 93. $89647.86 X 4, 2, s|. 
94 to 96. $79943.52 X , 6, 71. 

Amount, $3327494.19. 

97 to 100. If a steamboat is worth $3456, what will | be 
worth? What f ? What f ? What f ? Total, $6912. 

REMARK. 2 fifths will cost 2 times as much as 1 fifth; 3 fifths, 
3 times as much as 1 fifth, etc. 

101 to 105. T V of $155367 is how much? -l=? |=? 
f=? |=? Total, $97720.910f 

37. 2^0 divide by 10, we point off one figure on the right, 
by 100, two figures, by 1000, three figures. Those on the 
left will be the quotient, those on the right the remainder. 
500 divided by 10=50.0. $500 divided by 100=$5.00. 

REASON. By pointing off one figure we remove all the figures 
one place further to the right, so that the tens stand where the units 
were, and are units, the hundreds where the tens were, and are tens. 

2. It will be observed that the number pointed off corresponds 
with the number of ciphers in the divisor. For 10 we point off one 
figure; for 100, two; for 1000, three. 

31456-^-100=314.56, or 314^ 

3. Observe, also, that a decimal fraction, as .65, is changed to a 
common fraction by removing the point and writing the figure 1, 
with as many ciphers annexed as there are figures in the decimal: 



106. 134567 10=? 

107. 34578 100=? 

108. 713641000=? 

109. 21789 100=? 

110. 2163 100=? 

111. 78651000=? 



112. 
113. 
114. 
115. 
116. 



9876 10=? 
3453100=? 
6543 10=? 
2197100=? 
1763 10=? 



117. 7974100=? 



Total quotients, 14121.229 and 1954.44. 
38. To divide dollars and cents, the decimal point is re- 
moved to the left, which is the same as pointing off. 
6 



66 NELSON'S COMMON-SCHOOL ARITHMETIC. 

To divide by 10, it is removed one figure; by 100, two 
figures. $55.10 divided by 10=$5.510. $167.56 divided 
by 100=81.67,56.* 
Divide the following: 

118. $457.87-4- 10. 122. $473.041000 and 100. 

119. $1677.45-4- 100. 123. $15.17 10 and 100. 

120. $6109.88^-1000. 124. $16.57 100 and 10. 

121. $14999.99-4- 100. 125. $106.07-4- 100 and 1000. 
Total, $218.66. Total, $9.85,9 

126. Divide the following sums of money by 100: $645, 
$1678.25, $87493.57, $16453.27, $1998.38, $643.24, $2168, 
$4137.54. Total answer, $1152.16,9. 

39. It often happens that there are not as many figures 
to cut off as there are ciphers in the divisor. In such 
cases we prefix ciphers to the dividend to make up the 
number. 

Divide $5. by 100. Ans. .05. 

EXPLANATION. This is the same as removing the decimal point 
two places to the left, as above. The $5 had the decimal point on 
the right of the 5; it is now two places further to the left, and 
therefore is divided by 100. The cipher, in this case, as elsewhere, 
possesses no value. 

127. $5-4- 10=.5 132. $0.03-- 10=? 

128. $3^- 100=.03 133. $0.02-4-100=? 

129. $4-f-1000=.004 134. $0.14-=-100=? 

130. $50-4-1000=.05 135. $3.16-f-100=? 

131. $457-4-1000^.457 136. $21.30-4- 10=? 

Total, 1.041. Total, 2.1662. 



*NOTE. The value of each and all of the figures decreases ten- 
fold for every figure the decimal point is removed to the left. 

The $5 in the first example became 50 cents and the 10 cents be- 
came 10 mills or 1 cent, making the answer 5 dollars, 51 cents, not 
6 dollars, 510 cents. The second answer is 7 dollars, 67 cents, 5 
mills and / of a mill, or $1.07,5^. 



^~. \ 



DIVISION. 67 

137. Divide the following sums by 100: 3 cents, 33 
cents, $3.33, $33.33, $333.33, $3333.33. 

Total, $37.03,68. 

40. To divide by 20, 300, 5000, etc., we point off as 
many figures in the dividend as there are ciphers in the 
divisor, and divide by the 2, 3, 5, etc. The figures 
pointed off will form part of the remainder.* 

138. Divide 317745 by 500. 5 1 00)3177 1 45 

EXPLANATION Pointing off two figures, we 635245 

divide by 100; what is left we divide by 5. 635 ?~ 

139. 467831-4- 20=23391^1 142. 716849-=-700=? 

140. 71 6893^-300=2389^9 3 143. 897653--900=? 

141. 417368--500= 834|g 144. 49673--- 80=? 

Total quotients, 2641. Total rein., 1573. 
The answers to the following are required in dollars, 
cents and mills, omitting the remainders: 

$2131.51500=? Reduced to mills, it is 2141510. 
5 | 00)21315 | 10 

4263 ^=$4.26,3^. 

145. $13764.75-4-50=? 149. $16789.37-- 80=? 

146. $73968.23-4-60=? 150. $67859.67-^-900=? 

147. $37437.18^-90=? 151. $54168.23^-700=? 

148. $18964.20-4-80=? ' 152. $78910.00-4-600=? 

Total, $2161.11,8. Total, $494.16,5. 

153 to 157. Divide the following sums by 20, and give 

the answers as above: $1367.25, $3143.57, $2345.87, 

$34.57, $45679.44. Total, 2628.53,3. 

*When dividing dollars and cents, reduce them by erasing the 
decimal point and annexing ciphers if necessary. 
$357/27 ~500~ ? 
Reduced to cents the dividend is 35727. 5 | 00)357 | 27 

~Kffi 



68 NELSON'S COMMON-SCHOOL ARITHMETIC. 

158 to 165. Divide $34567.25 by 10, 12, 20, 100, 30, 
50, 70 arid 90. Total, 11132.84,6. 

166 to 176. Divide $367897.87 by 100, and the quotient 
by 10, 20, 30, 40, 50, 60, 70, 80, 90. Total, $4719.74,4. 

177 to 187. Divide $17654.37 by 100, and the quotient 
by 3, 10, 7, 40, 30, 50, 70, 90 and 80. Total, $298.78. 

188 to 196. Divide $314937 by 100, and multiply the 
quotient by 7; then divide the quotient by 30, 60, 40, 
12, 9, 80, 90. Total, $22968.52,7. 



VI. PERCENTAGE.* 

41. Percentage is the method of reckoning by Jiun- 
dredths. 1 per cent, is the one hundredth part, 2 per cent, 
twice that amount, 3 per cent, three times that amount. 

42- The sign is %. 25% signifies 25 per cent. 

43. To compute percentage, or, in other words, to find 
any rate per cent., we first find one per cent., and multi- 
ply it by the given rate. 

1. To find 5% of 350. 

One per cent, is the number divided by 100 or 3.50. 
5 per cent is 3.50x5=17.50 or 17^% or l7|i 

2. Find 6% of 3572. 

OPERATION. 35.72=1 per cent. 

6 



214.32 or 2U^=6%. 

3. 6% of 3146 is how much? Ans. 188.76 or 

4. 5% of 1937 is how much? Ans. 

*This rule is of such general utility, is so simple in its appli- 
cation, and so strictly belongs to the subject of division, that I can 
not refrain from introducing it in this place. 



PERCENTAGE. 



5. 7% of 3176 is how much? 

6. 9% of 7854 is how much? 

7. 8% of 396 is how much? 

8. 4% of 243 is how much? 

Amount, 970^. 

13. 20% of 3161=? 

14. 33% of 798=? 

15. 55% of 654=? 

16. 19% of 321=? 

Amount, 1316^. 

21. 12J% of 167=? 

22. 151% of 364=? 

23. 371% of 910=? 

24. 50J% of 693=? 

Amount, 7< 



9. 4% of 1300=? 

10. 6% of 367=? 

11. 9% of 463=? 

12. 8% of 6735=? 

Amount, 654.49. 

17. 30 % of 4541=? 

18. 23 % of 147=? 

19. 60 % of 7163=? 

20. 33% of 4371=? 

Amount, 7150^. 

25. 2% of $320=? 

26. 3% of $976=? 

27. 5% of $8900=? 

28. 7% of $6540=? 

Amount, $938.48. 

33. 20 % of $1361=? 

34. 451% of $316=? 

35. 17 % of $2163=? 

36. 19J% of $1723=? 

Amount, 1119.67,5. 

37. 25% of $264.50 is how much? 

2.6450=1% 
J 25 

132250 

52900 



29. 7% of $327=? 

30. 9% of $100=? 

31. 12% of $978=? 

32. 25% of $179=? 

Amount, $194.00. 



66.1250 or $66.12,5=25%. 



38. 3% 

39. 5% 

40. 6% 

41. 7% 



of $674.75=? 43. 

of $198.45=? 44. 

of $786.70=? 45. 

of $14.13=? 46. 

42. 12% of $1.19=? 47. 
Amount, $78.49,8 T V 



6% 
12% 
11% 
40% 
50% 



of $397.25=? 
of $187.17=? 
of $710.00=? 
of $1678.00=? 
of $7764.82=? 



Amount, 4678.00,5 T 4 n . 



70 NELSON'S COMMON-SCHOOL ARITHMETIC. 



48. 37J% of $461.75=? 51. \*\% of $610.18=? 

49. %\\% of $198.18=? 52. \\% of $114.14=? 

50. 62%% of $213.07=? *Amount, 539.60,8 r 6 5 ff . 

GENERAL EXERCISES. 

1. Divide $367.22 equally among 7 persons. Each will 
have ^; divide by 7. 

2. 12 horses, of equal value, cost $3456.24; what was 
the cost of each? 

3. If 5 men accomplish a piece of work in 320 days, 
how long will it take one man to do it? 

4. In a year there are 365 days; how many weeks will 
that make? 

5. A ship sails 84 miles in 12 hours; what is her aver- 
age speed per hour? 

6. How many shillings in 13456 pence? how many 
pounds? 

7. If 30 bricklayers can erect the walls of a house in 
120 days, how long will it take 12 to do it? 

8. How long will it take a writer to copy a speech of 
22340 words, if he writes 40 words a minute ? 

9. If $345.72 be divided among 12 persons, how much 
will each receive? 

10. In a pound there are 20 shillings; how many pounds 
are there in 3456 shillings? 

11. How many shillings are there in 21345 pence? how 
many pounds? 

12. How many bushels of wheat in 134563 pounds, 
reckoning 60 pounds to the bushel? 

Answers: 1600, $52.46, $288.02, 52}, 2242f, 7, 16, 
88 18 9, 9, 172 16, 300, 5581 $28.81, 1121 T 4 2 , 1778 9. 

*The Teacher should give numerous oral exercises on this rule, 
else scholars will be apt to err in pointing the products. 



GENERAL EXERCISES, 71 

13. A and B ar-e in partnership, in which A invests 
$30 and B 20. They make 14; how much should each 
receive ? 

A's share is 30 parts. 
J3 $ share is 20 parts. 

Together they have 50 parts. 

1 part of $14 divided into 50 parts=28 cents; and 20 parts= 
$5.60, nnd 30 partsr=r$8.40, which, added together $14. 

14. 2 men trading horses put in each $1200 and $800, 
and gained $1250; what was each man's share? 

15. If 20 men do a piece of work in 30 days, how long 
will it take 1 man to do it? how long 15? 

16. In 3683 oranges how many dozen? 

17. If $3678.21 be divided between 7 persons, how 
much will each receive? 

18. The profits of a speculation in which was invested, 
by 3 persons, $300, $900 and $800, are $919.18; how much 
should each receive? 

19. The cost of muslin is 30 cents per yard; how many 
yards can be bought for $397 at that rate? 

20. Find the cost of the following articles: 125 Ibs. 
sugar at 27 cents a pound; 37 Ibs. of butter at 37Jc; 2 
hams, each weighing 13 and 14 Ibs., at 21c; 115 Ibs. of 
cheese at 15J c. 

21. How much money will buy 37 \ Ibs. of tea at $2 a 
pound; 150 Ibs. of fish at 8Jc; 57 Ibs. of sugar at 31Jc; 
56J Ibs. of lard at 16 c ; 45 Ibs. of soap at 7J c; 31 Ibs. of 
candles at 16 c? 

22. A lady bought the following goods and paid for 
them out of $100; how much had she left? 8J yds. of 
French merino @ $2.25; 2 pieces brown muslin, 33 ard 
50J yds., at 35 c; 1 bonnet at $7.50; 1 shawl at- $11.50. 
- Answers : 750, 500, 600, 40, 306f , $23.73,7,* , $33.11, 
$525.45f, 1323& $71.12, $367.67,2, $137.87;T $413.63,1. 



'72 NELSON'S COMMON-SCHOOL ARITHMETIC. 



VII. LONG DIVISION* 

THE previous operations in division were performed al- 
most mentally, the learner writing only the quotients. 
That method is preferable when the divisor is found in 
the tables, or can be reduced to a number contained in 
them, as 500, 1200; otherwise the operation would be too 
difficult and tedious to perform mentally. 

44. Long Division has then to be used, which consists 
in writing the products and remainders as well as the 
quotients. The better to illustrate this method, an ex- 
ample which can be solved by short division is selected. 

1. Divide 3147 by 6. 

Divr. Dii-d. Quot. EXPLANATION. 1. To perform this operation, 
o)ol47(5^4g we try, as before, how often the divisor is con- 
tained in part of the dividend. 6 is contained 
14 in 31, 5 times, and writing 5 in the quotient, we 

12 multiply the divisor by it and write the product 

(30) under 31. 

2. We now subtract the 30 from the 31 as we 
v/r 

would perform an operation in subtraction. The 

remainder is 1. Instead of supposing this 1 to 
stand before the 4 in the dividend, we bring down the 4 to it, mak- 
ing 14. 

3. 6 in 14 is contained 2 times, which 2 we write in the quotient, 
multiply it upon the divisor (6) and write the product underneath. 

4. Subtracting this 12 from the 14, we have a remainder of 2, to 
which we annex 7 from the dividend, making 27. 

5. 6 in 27, 4 times. Writing the 4 in the quotient, we multiply 
it upon 6, making 24, which we write underneath. 

6. Subtracting, as before, we find a remainder of 3, and as there 

*This, considered the most difficult rule in arithmetic, may be 
deferred until the learner has passed Easy Fractions. 



LONG DIVISION. 73 

are no more figures in the dividend to bring down, we consider this 
the final remainder and express it in fractional form, as in short 
division, |. 

To enable the learner to comprehend this part of divis- 
ion more clearly, another example is introduced. 

3. Divide 8317517 by 723. 
723)8317517(11504} ~ 5 1 The product of the divisor (723) and 

1 723* ! I of the quotient. 

2 Tno'r 2 The remain der of 831723, with 7 
! annexed. 

*^ 3 The product of 723 and the second 

4 3645 figure of the quotient. 

6 3615 4 The remainder of 1087723, with 6 

6 orj-j n annexed. 

7 2892 5 The P roduct of 723 X 5 ) tne third fig- 

ure of the quotient. 

8 125 6 The remainder, with two figures (1 

and 7) annexed. 723 was not contained in 301, so another figure 
was annexed. 

7 The product of 723X4, the fifth figure of the quotient. 

8 The remainder. This is represented in fractional form in the 
quotient. 

REMARKS. 1. Instead of using the whole divisor in finding a 
quotient figure, it will generally do to use only the first one or two 
figures. In the preceding exercise, the first figure alone (7) was 
used, and in this way: "7 is contained in 8 how many times?" 

2. The products should never exceed the numbers above them. 
Number 3 should not exceed number 2. If on trial, it is found they 
do so, then a smaller number should be put in the quotient. 

3. For every figure brought down from the dividend, there should 
be one in the quotient. When the divisor is not contained in the 
new dividend, a cipher should be placed in the quotient and another 
figure annexed. 

4. The divisor can not be contained more than 9 times in the 
smaller dividends, as 1087, 3645. 

* The learner should put a mark under each figure brought down 
to prevent its being taken twice. 

T 



74 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



4. 71036- 
5. 31978- 
6. 167864- 
7. 9765837- 
8. 1763- 


- 21= 3382^4 
- 43=: 743f| 
- 54= 3108ff 
- 65=150243|f 

- 76=? 


13. 3167- 
14. 71438- 

15. 67898- 
16. 78637- 
17. 1000Q- 


- 129=? 
- 320=? 
- 764=? 

- 892=? 
-7109=? 


9. 7964- 


- 87=? 


18. 7185- 


-1990=? 


0. 89737- 


- 98=? 


19. 67416- 


- 144=? 


1. 77168- 


- 19=? 


20. 3784- 


- 642=? 


2. 3167- 


-119=? 


21. 14098- 


- 671=? 


52. *$730.45-r-126=? 25 


. $89289.61295=? 


53. $164.87-i-144=? 26 


$21008.97444=? 



24. $1710.14-^166=? 27. $10000.00180=? 

Answers: 8 to 12, inclusive, quotients, 5116; remain- 
ders, 211. 13 to 16, quotients, 423 and 956. 17 to 21, 
498 and 19246. 22 to 24, $39.54; remainder, 6. 25 to 
27, $405.53; remainder, 629. 

When there are ciphers in the divisor, they may be 
pointed off with a corresponding number of figures in the 
dividend. 

28. 67314968-^-163000 is how much? 

163 1 000)67314 1 968(412}|f -g 
652 



211 
163 



The figures pointed off in the dividend 
were annexed to the remainder, forming the 
484 fraction, jff |gg. 
326 



158 

29. 12986745 7300=? 

30. 8109867018000=? 

31. 513643 2500=? 
Quot's, 6489; Kern., 9858, 



32. 7613412-- 37100=? 

33. 4567800-;- 20900=? 

34. 5632710-^-171000=? 
Quot's, 455; Hem.; 180222. 



* Express the dollars in cents before dividing. 



LONG DIVISION. 75 

35. Find the sum, difference, product and quotient of 
128097 and 8070; of 1736009 and 4761 ; and 4070391 and 
71068, omitting the remainders. 

Totals, 289282688427, 1033998999, 8268611231 and 
29993710999. 

PRINCIPLES OF DIVISION. 

45. If we divide the price of a number of things of 
equal value by the number, we obtain the price of one. 

46. The quotient will usually be in the same name with 
the dividend or number to be divided. If the dividend 
be dollars, the quotient will be dollars ; if it be rods, the 
quotient will be rods. 

36. If 75 barrels of flour cost $450, what was the price 
per barrel? 

37. If 125 horses cost $25000, what was the cost of 
each? 

38. If $167809 be divided among 7614 persons, how 
much should each receive? 

39. How much tax should each of 16785 persons pay 
of a levy of $71683? 

40. In 303,656,837 Ibs. of cotton, how many bales, sup- 
posing each bale to weigh 320 Ibs.? 

41. If $79640 be divided among 274 persons, how much 
will each get? 

42. The earth moves round the sun at the rate of 
66600 miles an hour; at what rate does it move per 
minute? 

43." If 357 yards of broadcloth cost $1035.30, what was 
the cost per yard? and what would be the cost of 50 
yards at the same rate? 

Answers arranged promiscuously: 1110, 948, 
$290.65, $6, $22.30,9, $145, $4.27,7, $200, $2.90. 



76 NELSON'S COMMON-SCHOOL ARITHMETIC. 



EXERCISES IN MULTIPLICATION AND DIVISION.* 

1. If 23 yds. of inuslin cost $3.45, what will one yard 
cost? 

2. If 117 men can do a piece of work in 48 days, how 
long will it take 3 times that number to do it? 

3. How many men can do a piece of work in 5 days, 
that took 10 men 25 days? 

4. If a case holds 29 pieces of muslin, how many will 
it take to hold 7250 pieces? 

5. If 15 men can do a certain piece of work in 75 
days, how long will it take 1 man to do it? 

6. If 7 dozen silver spoons cost $35.35, what will 3 
dozen cost? 

Find the cost of one dozen, then the cost of 3. 

7. If f of a ship cost $14602, what will the ^, or the 
whole ship cost? 

8. If | of a piece of property cost $6377, what will -J 
of it cost? 

9. In a cord of wood there are 128 feet, how many 
cords are in a pile measuring 4 feet wide, 8 feet deep and 
100 feet long? 

10. In an acre there are 4840 square yards: how many 
are there in -J- of an acre ? 

11. A field contains, 12 acres, and is 660 yards long; 
how many yards is it in breadth? 

12. A tract of 5 acres is 220 yards long; how much 
should be cut off the breadth to leave 1 acre? 

13. -J- of a dozen books cost $7.50; what was the cost 
per dozen? 

14. of a dozen cost 22 cents; what was the cost per 
dozen? 

*For answers, see next page. 



PROPERTIES OF NUMBERS. 77 

15. 1^ dozen cost $1.20; what was the cost per dozen? 

16. T 7~ of a hundred cost $28; what was the cost per 
hundred? 

17. 2 3 jy of a hundred cost |15.75; what was the cost per 
hundred ? 

18. 1 1 of a hundred cost $3.15 ; what was the price per 
hundred? 

Answers: 51107, 3644, 25, 16, 50, 250, 22, 15.15, 605, 
60, 88, 80, 33, 100, 4.20, 105, 1125, 15, 88. 

47. METHOD OF PROOF. Division and Multiplication 
beiDg converse operations, the one is proved by the other. 

DIVISION. PROOF. 

38)3715(97 97=quotient. 

342 3S=divisor. 

~295 776 

266 291 



29 rem. 3686+tfie rem., (29)=37lb=dividend. 

MULTIPLICATION. PROOF. 

multiplier, product, multiplicand. 

465 25)11625(465 

_25 100 

2325 162 

930 150 



11625 125 

125 



VIII. PROPERTIES OF NUMBERS. 

48. An Integer is any number considered as a whole, 
as 3, 7. 58, 129. 

49. A Fraction is a part of any thing or number of 
things considered as a whole; ^, -?-, -f-fc. 

50. Numbers are divided into Odd and Even. 



78 NELSON'S COMMON-SCHOOL ARITHMETIC. 

An Odd number can not be divided into two equal parts 
without a remainder, as 1, 5, 57. 

An Even number can be divided into two equal parts 
without a remainder, as 4, 10, 68. 

51. Numbers are either Prime or Composite, Abstract 
or Concrete. 

52. A Prime number is an original number, or one 
which can not be produced by multiplying two other 
numbers together, as 1, 7, 31. 

53. A Composite number is one which may be com- 
posed of two other numbers multiplied together, as 8, 
which is composed of 2 and 4 multiplied together; and 
27, which is composed of 9 and 3 multiplied together.* 

Exercise. Write out 50 prime and 50 composite numbers. 

54. An Abstract number is an unapplied number, or 
one which conveys the idea of number exclusively, as 4, 
15, 47. 

55. A Concrete number is an applied number, or one 
which conveys the idea of something else besides number. 
The above numbers become concrete when applied as fol- 
lows : 4 mills, 15 dollars, 47 pounds, and the names, 
mills, dollars and pounds, are called denominations. 

56. A Multiple is a number which contains another 
number a certain number of times without a remainder. 
12 is a multiple of 3 as well as of 2, 4 and 6. 

57. A Common Multiple is one which contains two or 
more numbers a certain number of times without a re- 
mainder. 12 is a common multiple of 2, 3, and 4. 

58. The Least Common Multiple is the least number 



*NOTE. Since all even numbers are divisible by 2, an even 
number can not be a prime number, nor can any number ending 
with 5; it follows, therefore, that every prime number, except 2 
arid 5, ends with 1, 3, 7 or 9. 



PROPERTIES OF NUMBERS. 79 

which will contain two or more numbers without a re- 
mainder. 6 is the least common multiple of 3 and 2, and 
18 of 3 and 6, and 24 of 2, 3 and 8. 

59. An Aliquot is a number which will divide another 
without a remainder. The parts of which a multiple is 
composed are called aliquot parts of that number. 1, 2, 
3, 4 and 6 are aliquot parts of 12. 

60. Complement: The number required to be added 
to another to make it equal to a larger. It is usually ap- 
plied to 100, 1000 or some other power of 10. Taking 
87 as a part of 100, the complement is 13, or 50 as a 
part of 60, the complement is 10. 

Exercise. Taking 35 as a part of 60, 70 as a part of 
100, 18 as a part of 20, 73 as a part of 80, required the 
complements. 

61. Even numbers are divisible by 2 without a re- 
mainder. 

62. If the two right-hand figures of a number are di- 
visible by 4 without a remainder, the whole nun^ber is 
also divisible by 4. 

63. Numbers ending with 5 or are divisible by 5 
without a remainder. 

64. If the three right-hand figures of a number are di- 
visible by 8 without a remainder, the whole number is 
divisible by 8. 

65. If the sum of the figures of any number is divisible 
by 3 or 9 without a remainder, the whole number will be 
divisible by 3 or 9. 

MULTIPLICATION BY ALIQUOTS. 

66. To multiply by 2-i, it will shorten the operation if 
we multiply by 10, which is 4 times too much, and then 
divide by 4. In the same way we can multiply by any 
other aliquot of 10, or by aliquots of 100, 1000, etc. 



80 NELSON'S COMMON-SCHOOL ARITHMETIC. 

To multiply 176 by 12|. 

8^17600 EXPLANATION: 176 being multiplied by 100 is 

. - 8 times more than the sum required, so we divide 

2200 Ans. by 8> 

To multiply 379 by 250. 
4)379000 379 being mu itiplied by 1000 is 4 times too 

94750 Ans. much, so we divide by 4. 
To multiply $49.75 by 125. 

8)4975 000 The $49.75 are considered as cents and mul- 

/>oi o-re tiplied as the preceding. 

621 8*5 cents. *! .. . ..,. 

A/irt-m *r A REMARK. It will lessen the work still more 

or $6218.75 AM. . . , * 14 . .. , 

to simply assume the number to be multiplied 

by 10, 100 or 1000. 

ALIQUOT PARTS OF 10, 100, 1000. 

To be committed to memory. 

ALIQUOTS OF 10. ALIQUOTS OF 100. ALIQUOTS OF 1000. 

5 =4 50 =4 142=4 333=i 

3i^i 331=4 12= 250=1 

2i=3 25 =i 10 = T ^ 166^ I 

2 =1 20 =i 8^=^ 125 =^ 



The pupil can prove the accuracy of his calculations by 
multiplying in the ordinary way. 

4. 140X 12i=? 13. 949 x333i=? 

5. 3767 X 8|=? 14. 179 X 2-i=? 

6. 9987X 25^? 15. 769 X 3i=? 

7. 9174X125 =? 16. 12-' X 19 =? 

8. 3689X 33^ = ? 17. 125^X787 =? 

9. 9210X 16'j=? 18. 250 X125 =? 

10. 7897X166^=? 19. 16* X 48 =? 

11. 8997X 50 =? 20. 83^x756 =? 

12. 786X 14$=? 21. 197 X 12=? 



PROPERTIES OF NUMBERS. 81 

23. 675 yards @ 37^ cents. 

OPERATION. 675 at & dollar $675.00 

at 25 c=J "168/75" 
at 12^ c^J- 84.374 

^w. 253^121 

24. 715X62J cents. 31. 9876x$2.18f 

25. 947XS7A 32. 719x?3.62 

26. 194xl8f 33. 965x$4.37J 

27. 567X31} 34. 758x$l-25~ 

28. 619X374 35. 197x$2.87.V 

29. 1060X324 36. 879x$3.95~ 

30. 197X75" 37. 179XW.32J 

38. To find the cost where there are fractions in both 
factors: 18| Ibs. @ 12i cents. 

OPERATION. 18| Ibs. @ $1=$18.75 

at 12in=^ or $2.34| Arts. 

39. 37J Ibs. @ 18| cents. 

OPERATION. 37^ @ $1=837.50 

at 12^=-^ 4.687 
at 6^=4 2.343 

Am. $7.030 

40. 176^ doz. @ 18| cents. 41. 164J Ibs. @ $1.62J 
73l| doz. @ 12J 374J Ibs. @ 2.25 

doz. @ 37J 693J Ibs. @ 0.16f 

Amount, $197.06. Amount, $1224.93. ' 



E . The multiplier in the 25th Ex. wants only 12J cents, or J, 
of being a dollar; so we find the cost of 947 at a dollar and take 
off I of it. For 32J take 30, and 2J as J- of 10. 



ill 

/rf 



82 NELSON'S COMMON-SCHOOL ARITHMETIC. 



IX. EASY FRACTIONS.* 

67. A FRACTION is a part or number of parts of any 
thing considered as a whole. Fractions are of two kinds, 
common and decimal. A common fraction is written with 
two numbers, called terms, having a line between them, as 
j^; a decimal fraction with one number, having a period 
at the left, as .5 (five-tenths). 

68. A common fraction indicates division, the upper 
number being the dividend and the lower the divisor. In 
treating of fractions, the dividend is called the numerator 
and the divisor the denominator. 

The denominator indicates the number of parts into 
which the whole is divided, and the numerator the num- 
ber of such parts under consideration. 

69. VALUE OF A FRACTION. The lowest value of a 
fraction is expressed by the figure 1 for a numerator, and 
the highest value a number as great as the denominator 
less l.f ^ represents the lowest value of fractions of the 
denomination of ninths, while . | represents the highest 
value of that denomination. J 

*This chapter is introduced for the benefit of that large class of 
pcholars who leave school before completing the study of Arith- 
metic. The subject of fractions is treated of at length in the latter 
part of this book. 

|This does not apply to improper fractions, which, as the name 
indicates, are not strictly fractions. 

1. Since this is the case, it is evident that fractions decrease in 
value as their denominators increase, the numerators remaining the 
same. 1 is less than ^, 1 than 1, than 1. 

2. It is also evident that the value of a fraction depends on the 



EASY FRACTIONS. 83 

When a number is divided into two parts, each part is 
called a half ; into 3 parts, each part is called a third; 
into 4 parts, each part is called a fourth; into 5, a fifth; 
into 12, a twelfth; into 18, an eighteenth; into 25, a twen- 
ty-fifth; into 100, a hundredth; into 476, a four hundred 
and seventy-sixth part. 

ORAL EXERCISES. 

1. When a number is divided into 10 parts, what is 
each part called? Into 11? Into 20? Into 33? Into 
45? Into 97? Into 62? 

2. When divided into 31, what? Into 69? Into 103? 
Into 364? Into 155? Into 1000? Into 3144? 

3. Which is the greater fraction, ^ or J ? -| or J? 

o'i? TYr T ' s ? 2 'ar ,',? ,' s or &? 

4. Which is greater, f or |? Ans. |. 

REASON. Because it will take less to make it a whole number. 
The first fraction requires J to make it a whole number, while this 
one requires only J. 

5. Which is the greater, f or f ? |orf? |or|? f 
or I? f or I? ?or T y if or ji? f or ?? 

6. Which is the greater, if or ff ? ^f or f|? jf or 
H? i*orf? IforAf? 14 or if? 

Since the value of a fraction depends upon the relation 
of the numerator to the denominator, [note 2, page 82,] 
both terms may be multiplied or divided by the same 
number without altering its value. 

_4 ?^- 2 _ 1 

~ ~~ 



relation of the numerator to the denominator, or, in other words, 
the number of times the numerator is contained in the denomina- 
tor. | is equal to |, because the numerator 3 is contained in its 
denominator, 6, the same number of times that the numerator 4 ia 
contained in the denominator 8. 



4 NELSON'S COMMON-SCHOOL ARITHMETIC. 

Now, | and i possess the same value as f , because 
their respective numerators are contained the same num- 
ber of times in their denominators. 

EXERCISES FOR THE SLATE. 

5. Change | to twentieths. 

3X& 15 EXPLANATION. By multiplying the 4 by 5, we change 
4X& ^0 the denominator to twentieths; arid by multiplying the 
numerator by the same number we preserve the same value. 

1. Change -| to 8ths; to 12ths; 4 to 20ths; f to 
Uths; I to 12ths; f to 18ths; to BOths. 

2. Change ^ to 20ths; f to IGths; f to 27ths; j| to 
52ds; | to 25ths; fg to 150ths. 

3. Change | to 32ds; g to 40ths; Jf to 72ds; fg to 
104ths. 

4. Change to SOths; if to 52ds; || to 128ths; T 4 
to 50ths. 

5. Change T ^ to 6ths ; ^ to 5ths; T 8 2 to 4ths; j| to 
8ths; if to 7ths; j to 8ths. 

6. Change | to halves; T 6 a to 5ths; T 4 ^ to 4ths; || to 
8ths; fg to 3ds. 

The fractions in last exercise, (6th,) when changed as 
required,' would be reduced to their lowest terms; that is, 
expressed in their simplest form. 

70. To reduce a fraction to its lowest terms is, there- 
fore, to divide the numerator and denominator by such a 
number or numbers as will do so without a remainder. 
When the terms can not be divided exactly by any num- 
ber greater than 1, the fraction is in its simplest form. 

Reduce T 6 2 to its lowest terms. 



T y, T? , , T | g , ^ -fa to their lowest terms, 
Answers: , , T ' r , &, i, 4, ^. 



EASY FRACTIONS. 85 

When a single number will not reduce the fraction, other 
numbers may be used, as below. 
2. Reduce --f f-g- to its lowest terms. 



3. Reduce to their lowest terms, -%, -f^, -J|f, and 

4. Reduce to their lowest terms, -$> TT^ sWb^ 

T5~3T' 

5. Reduce to their lowest terms, ^ 7 e 4 7 %, ^Irolb anc ^ 
Answers : ff, ffr, A. H- A> A. Mi. f, T&> #7, 

TTHF5 ITT' "8"' 

Fractions may be Proper, Improper, Simple, Compound 
or Complex. We shall treat of only the three former at 
present. 

A proper fraction is one whose numerator is less than 
its denominator, as ^. An improper fraction is one whose 
numerator is equal to or greater than its denominator, 

nc 5 P 

as TJ, TJ. 

71. A simple fraction is a single fraction, and may be 
proper or improper, as |, |. 

72. When a whole number and fraction appear together, 
they are called a mixed number, as 5|. 

73. Improper fractions may be changed to whole or mixed 
numbers by dividing the numerator by the denominator.* 

To change - 1 /- to a mixed number. 

5)13 EXPLANATION. There are 5 fifths in one whole num- 
~T 3 ber; in 13 fifths there are as many Is as the number of 
5 times 5 is contained in 13, which is two times, with 3 
fifths over, making 2^. 

*This is simply acting on the principle that the numerator is the 
dividend and the denominator the divisor. 



86 NELSON'S COMMON-SCHOOL ARITHMETIC. 



1. Change |, g, |, J g &, 2.^-L, *. to whole or mixed num- 
bers. 

2. Change the following : -*/, ^_, *-, - 2 T 2 -, ^-, -^-, 1. 

3. Change i-f o, ^-, -^ ff, 44,1, ^, A J a - 
Answers: lj, If, 3, H, 301, 691, 53,. 8, 811, 7, 58^, 

56f, 184, 13|, 3^, 81, Hi, 4i, 91, 18f . 

74. To change whole or mixed numbers to improper frac- 
tions is an operation the reverse of the last, which, scarcely 
needs explanation. 

4. Change 94 to an improper fraction. 

9i EXPLANATION. In 1 whole number there are 5 fifths; 
5 in 9 there are 9 times 5 or 5 times 9 fifths, to which we 
"JjJ add 4 fifths, and we have 4 ^. 

5 

5. * Change the following mixed numbers to improper 
fractions: 3|, 9|, 8?, 5|, 41f, 97J-, 16|. 

6. Change the following: 7|, 9}J, 4|, 7 If, 18|, 16, 
124. 

7. Change the following: 21 to fifths; 16| to eighths; 
121 to fourths; 16 to twelfths; 8 to twelfths; 131 to 
sixteenths. 

75. To multiply a fraction by a whole number is simply 
to multiply the numerator without altering the denomi- 
nator, or to divide the denominator without altering the 
numerator. 

To multiply T 7 2 by 6. 



REASON. Assuming that 7 is a whole number, multiplying it by 
6 gives 42; but since it is not a whole number, but twelfths, the 42 
is =8- or 3. 



*The learner should prove the accuracy of his work by last ar- 
ticle. 



EASY FRACTIONS. 87 

2- 6)/ 3 (.J=3J. 

By decreasing the denominator, the fraction is in- 
creased (as it takes fewer of the small parts to make a 
whole number) ; hence, the 7 represents halves instead of 
twelfths. =3J. 

1. | X7=? 5. J> u xl2=? 9. i*xll=? 

2. I X9=? 6. f|x 6? 10. f|Xl2=? 

3. 5x8=? 7. fix 5-=? 11. Jfx 8=? 

4. T * B X4=? 8. i|XH=? 12. X21=? 
Answers: l-, 6|, 5, 5f, 10|, 9&, 7|, 4ft, 4j, 6, 1||, 

12|, 6. 

76. To multiply a whole number by a fraction, we mul- 
tiply the numerator without altering the denominator. 

13. Multiply 25 by |. 

25X3 fourths=75 fourths, or 7 ? 5 , which, changed to a mixed 
number, [Art. 73]=18j. 

14. 35X|^-? 18. 134X 2 1 o-=? 22. 16X T 3 o= ? 

15. 21xf=? 19. 215X1 ? 23. 21X|=? 

16. 18X^=? 20.112X|=? 24. 

17. 116XA=? 21. 36X|=? 25. 

Answers: 28, 6/ ff , 4|, 174, 95|, 18, 3|, 931, 12, i|, 20, 
34|, 30|, 

77. To multiply a mixed number by a whole number. 
Multiply 7| by 9. 

4 EXPLANATION. 3 fourths multiplied by 9=27 fourths, 

9 or 6|; and the 7 multiplied by nme=63, plus the 669, 
69^ making the product 69|. 

Or thus: 7| 



The Teacher will find it important to require the learner to preserve 
the process, as he will be apt to adopt clumsy methods of solution. 



88 NELSON'S COMMON-SCHOOL ARITHMETIC. 

2i?. 1SJX 5=? 30. 29JX 8=? 34. S3 i X 7=? 

27. :>:\\ 8=? 31. IG-x 9= ? 35. 12^ X 8=? 

28. 12-.i\12^=? :>- 8t|xl2=? 36. 5.,- g X 9? 
21). :^\ 9=? 33. 62JX 9=? 37. 187 XH=? 

Answers: 3!, 93J, 3tf 62, 45 ,V 96 

: -. i;>>. 1050, i;>o. ;u>o." 

78. ^' multiply a whole number by a jni.ird number. 
38. Multiply 29 by 8|. 

29 

S; FMM. vNvriox. Multiplying -9 by i! thirds, we hare 

TIT 58 thirxls. or l^J, which we write in the first line. Then 
*Z* 29X8=232, which, added to 1%=251 j. 

Or thus : 29X-V= 1 3 d =- 51 ] 

15X 3]=? 42. 12\12i^? 4. r i. 14xl7f=zT 

40 27X 6|=? 43.47X37^=? 46. 29x18 1 =? 

41. ? 44. 93X16J^? 47. 83x 6 T " 

Ai> ' : . L-2J. 543|, 

522^ 150. 17t52^." 

To multiply a fraction by a fraction. 

4$. Multiply j by j. 

Assuming the numerator 5 to be ft whole number, jx^-^^ 1 - 5 : 
but 5 is not ft whole number, but 5 sixths; hence y is 6 timt* too 
MK& tj* dirided by 6z=|| T or | [Note l r p*ge 82.] 

79. Hence, to multiply & fraction by a fraction, we 
iHufcyrfjr the tttraierafer* together for a new numerator, and 
the denominators for a new denominator. 

? which, reduced to its lowest 



*It will be obserred that to mnltiplj by ft fraction does not m- 
II Mini Ike ultipli^^ as in whole numbers; bat, on the contrary, 
Aerates it, the f being less ihan J. 

To account for this, U Ls only necessary to remember that ft whole 



EASY FRACTIONS. 
49. -| X 2=? 52. X= ? 

5J. !;!xW ? 54 - *Xj=? 57. g/. ?=? 

Answers: f, , J, f, T ^, 3$, J, f, |, ,', 

80. ^ multiply a mixed number by a fraction or a miw.d 
number. 

58. Multiply 15| by T V 

15|=^, which, multiplied by f =* 4 J- or 14 4 7 . 

59. Multiply 8} by Ifif. 
8|=3/t an(1 lf; 3=^- 5 4 4 X^ 

60. 12^/lf^ -='! W>. 14'|X i 9 =? 66. 

61. 8JX2%=? 64. 23^X i=? 67. 

62. 37^X52*=? 65. |Xl4^^? 68. 

Answers: 126J, 35 , 231 J, 6 T 3 ^, ^, 208^, 245g, 29f, 
1978f 

81. To divide a wlwle number by a fraction or a mixed 
number. 

1. Divide 315 by |, or, in other words, find how often 
| is contained in 315. 

.SOLUTION. Before we can measure 315 by fourths, we must 
change it to fourths. In 1 there are 4 fourths; in 315 there are 
315 times 4 or 1260 fourths, which, divided by 3^420. Hence, \ is 
t contained in 315 420 times. 

OPERATION. 315 or 
4 

3)1260 
430 

t number is reduced to the denomination of a fraction by being mul- 
tiplied by it. 6Xf =18 fifths or 3|. Much more is a fraction re- 
duced in value if multiplied by a fraction. From this we readily 

infer, 

2. That to divide by a fraction increases the dividend. 

8 



90 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



136-5-?=? 



3. 27-f-J=? 5. 684-f- T =? 7. 

Answers: 32|, 365|, 24341, 19740, 2280, 158f, 24. 

8. Divide 25 by 5 J. 

OPERATION. 25x2 halves 5 ^ and 5|X2 ty. 50-f- 

82. Hence, to divide by a fraction, we multiply by the 
denominator and divide by the numerator, or invert the 
divisor and proceed as in multiplication. 

9. 157-4- 3|=? 12. 345-4- 6J=? 15. 195-=-16f=? 

10. 22-5-12=? 13. 39-4-15I-? 16. 39-5-12f=? 

11. 16-5-16|=? 14. 79-4-37^? 17. 87-4-3l|=? 
To divide one fraction by another. 

18. Divide f by |. OPERATION. f Xf ig T 9 o 
EXPLANATION. By inverting the divisor, we obtain ^g, the terms 

of which, being divided by 2, give T 9 . 

19. 9 _!_!_-? 22. 

20. /,-*-=? 23. 

21. 2_j_6 = ? 24. 

Answers : 1|, ff , f|, 38|f, 36|, 

83. To divide when either divisor or dividend is a mixed 
number and the other term a whole number, both terms may 
be reduced to the same denomination. [Art. 81.] 

28. Divide 3457] by 13. 

EXPLANATION. The dividend containing the 
fraction of ^, both terms are reduced to fourths, 
and division performed as in whole numbers. 
The result shows that the divisor is contained 
in the dividend 265 times, with a remainder of 
49 fourths [Art. 46], or 265|| times. 



-4-=? 25. -f-l= 



-4- =? 



. 

26. 
27. 



-4-6|=? 

f, f 



3457J 
4 

52)13829(265 
104 



342 
312 



309 
260 

49 



EASY FRACTIONS. 91 

The same by short division. 

13)3457 1 EXPLANATION. 13 is contained in 8457, 265 times, 

ope 4 9 with a remainder of 12, which, reduced to fourths, 

52 including the |- in the dividend, is 49 fourths. 13 

not being contained in this an even number of times, the denomi- 

nator is increased 13 times, (which is the same as to decrease the 

numerator,) which gives the same fraction as by long division, 4| 

29. 1398J 56 =? 35. 1255 | 350=? 

30. 256J-- 7 =? 36. 796 J 421=? 

31. 1939 -~ %=? 37. 467 f 12=? 

32. 7961?-f-300 =? 38. 214 1 9=? 

33. 9219 ~ 6J=? 39. 713^ 8=? 

34. 1391 -v- 56=? 40. 391Jt 6=? 
Answers: 24}g|, 24^, 1475&, 36 T \, 232^, 26ffg, 



84. To subtract a fraction from another of the same de- 
nomination is simply to subtract the less numerator from 
the greater. 

I. From T 7 a take T 3 . 

T 7 T 3 0=T 4 r I 

2- jf-H=? 5. T - T V=? 8. T g -^=? 

337 '25 _ ? (\ 1:2 _ 9 _ 9 Q 1 ( > 3 _ 9 

' 42 - 42 - ' ' 3T - 51 *' 3^ - 39 - r 

416 5 _ ? 7 16 7 _ 9 10 11 _ 6 _ 9 

55 - 52 - ' f' 45 - 45 - f 1U ' T7 IT - ' 

Answers: f, 4, J, T \, J, J, |, |, T 6 S , T \. 

85. ^> subtract a fraction or mixed nnmber from a whole 
number. 

II. From 9 take 3|. 

The following formula will render the operation simple : 

Whole number. Fourths. EXPLANATION. Arranging the less under 

the greater, we find we can not take 3 fourths 

_ from fourths; so a whole number or 1 is 

5 1 added to both terms. In 1 there are 4 fourths, 

QT 5i from which take 3 fourths, and we have a 

remainder of 1 fourth. To the 3 add 1 and 



92 NELSON'S COMMON-SCHOOL ARITHMETIC. 

we have 4, which, subtracted from 9, leaves 5, giving for the an- 
swer 5. 

12. 13 4J=? 15. 11 2 J=? 18. 52 2?4=? 

13. 15 5=? 16. 7 f =? 19. 13 12J=? 

14. 29121=? 17. 14 1 T 3 2 =? 20. 89 75J=? 

A n <JWAra 5 1 O5 J.1 94.6 Q2 Q3 11 1Q3 1Q3 

Answers. g , o g , J g , *^, *^, o 3 , p|, 10^, io ? , lo^. 

86. j?b subtract one fraction from another of a different 
denomination, it will be necessary first to reduce both to 
the same or a common denominator. 

21. From | take f 

By Art. 69, | can be changed to 56ths by multiplying 
both terms by 7, and f can be changed to 56ths by mul- 
tiplying both terms by 8, giving || and |-|, the difference 
between which is ^, the answer. 

It will be observed that the multipliers used in this case 
were the two denominators, 7 and 8, which, multiplied to- 
gether, give a common denominator, and multiplied into the 
numerators of each other give the new numerators. 
OPERATION. | =ff-4|= 5 i ff 

22. From f take i . 25. | 4=? 28. 

23. From f take \ . 26. 6J 1=? 29. 

24. From f take T 3 . 27. l| J= ? 30. T 3_ J_^? 
Answers: 2 5 , 5|, |f, |, T %, J, f|, |, ^, |. 

87 ^ -c?cZ fractions of the same denomination, the nu- 
merators only are added, and the sum reduced to a mixed 
number or its lowest terms. 

1. Add I+I+I+I- 

3 EXPLANATION. Here the four numerators are added 

6 together, making 21 eighths, which, reduced to a mixed 

5 number, are equal to 2. 






\ 



EASY FRACTIONS. 93 



2- 

3 31 4 1718 
9~r y ~r$ i 3 

4. 

5- 

6- 



88. ^> add fractions of different denominations, they 
should first be reduced to a common denominator, as iu 
subtraction. 

7. |+!-? |+|=|+|=Vt=lf or If 

When three or more fractions of different denomina- 
tions are to be added together, they may be reduced to a 
common denominator by multiplying all the denominators 
together, as above, and then by multiplying each numer- 
ator by all the denominators except its own.* 

8. Find the sum of i+f + 

2X^X6 48=: Common denominator. 
1 X 4 X 6==24=sJ'Wfcl numerator. 
3x2X^36 Second numerator. 
5x2x4 40 ^/ureZ numerator. 

Wfa^Stim of numerators. 
Hence, ^=2^=2^. 

The J in the example was multiplied by 24, giving ||; the f by 
12, giving || ; and the | by 8, giving |g. 

9. + =? 14. | + | + | -? 19. 2|+ | + | =? 
10. i + 2=? 15. i + | + i=? 20.5-i+6|+|=? 
11-T\+I= ? 16 - 7VH+.iV= ? 21. i+2|+ T 3 2 =? 

12. 2 + |=? 17. |+V 3 +'l=? 22. |+ + -^=? 

13. J + T V=? 18. | + I +/ 3 =? 23. ^+ ^+ J =? 
Answers: 2^, 1, 1^, Ii, **, i 12, 3|, 1|, 1 I/,, 






*This is simply multiplying both terms by the same number. 
[Art. 69.] 



94 NELSON'S COMMON-SCHOOL ARITHMETIC. 



X. THE MERCANTILE PROFESSION". 

THE mercantile community may be divided into various 
classes : Importers, Jobbers, Wholesale Dealers, Commis- 
sion, Forwarding and Retail Merchants, Brokers, etc. 

89. Importers purchase goods and produce in foreign 
countries, and sell them in the home market to jobbers 
and wholesale dealers. They also receive goods from 
abroad to sell on commission. 

90. Jobbers. This term was first applied to persons 
dealing in stocks to a limited extent, but it now includes 
nearly all classes of wholesale dealers. We speak of dry 
goods jobbers, produce jobbers, cattle jobbers, etc. 

91. Wholesale Merchants or dealers buy from import- 
ers, jobbers, manufacturers and producers, and sell to re- 
tail dealers. 

92. Commission Merchants* act as agents for other per- 
sons in buying and selling goods, collecting debts, etc., for 
which they charge a percentage on the whole amount of sale, 
purchase or collection. Merchants of this class usually keep 
a wholesale department in their warehouses, where they 
sell their own goods as well as those of others, and even 
ship merchandise to distant places for sale on commission, 
thus acting in the capacity of principals as well as agents. 

The person who sends goods to another to be sold on com- 
mission is called the shipper or consignor ; the person who 
receives them, agent, correspondent, consignee or factor ; and 
the goods or merchandise sent, shipment or consignment. 

93. Forwarding Commission Merchants and Express 
Companies are intrusted with the care of conveying goods 

* Auctioneers belong to this class. 



THE MERCANTILE PROFESSION, 95 

from one city or country to another, for which they charge 
a percentage, called forwarding commission. This class 
of merchants usually have warehouses, and on rivers, 
wharf-boats, for the storage of merchandise. A separate 
charge is made for all goods lodged in these places, ac- 
cording to the time they remain. The fee charged is 
called storage. 

94. Retail Merchants dispose of their goods in small 
quantities suited to the wants of consumers. 

95. Brokers form a numerous class. They assist com- 
mission merchants and dealers generally in finding buyers 
for their wares, and trade or speculate in stocks, lands, 
etc. By confining their attention to one line of business, 
they acquire a more intimate knowledge of its details and 
of the credit of persons engaged in it, and are thus pre- 
pared to render valuable services to both buyer and seller, 
between whom they act as middle-men. 

The business of the broker does not require the invest- 
ment of much capital, as, unlike commission merchants, 
they are not under the necessity of keeping stores or 
warehouses. For the same reason, their fees are smaller 
than those of the latter. 

There are Money, Exchange and Bill, Stock, Custom- 
house, Heal Estate and other Brokers. 

MONEY, EXCHANGE AND BILL BROKERS. 

96. They buy, sell and exchange specie, bills of ex- 
change, notes, etc. 

The entire business of this class is often performed by 
one individual or company. Being considered a branch 
of the banking business, many of them adopt the name 
of bankers. 

Heal Estate and Stock Brokers buy and sell for others, 



9G NELSON'S COMMON-SCHOOL ARITHMETIC. 

lands, houses, stocks in public funds and joint stock com- 
panies, etc. 

Custom-house Brokers find employment in maritime 
cities, by assisting masters of ships in obtaining the nec- 
essary papers at the custom-house, and paying duties or 
taxes incident to the navigation of the ocean. 

Ship and Insurance Brokers procure freights and car- 
goes for ships, adjust the terms of charter parties, settle 
accounts between owners and masters of ships, effect in- 
surance on ships and cargoes, etc. 

Produce Brokers buy and sell for others various kinds 
of farm produce, as corn, wheat, cheese, etc. They stand 
between the producer and dealer or shipper. 

The business of other brokers will be sufficiently indi- 
cated by their names. 

MERCANTILE AND COMMERCIAL COLLEGES 

are institutions of learning, having for their ostensible ob- 
ject to prepare young men for entering the mercantile 
profession. They are got up by private enterprise, some 
of them being chartered and some not. The chartered 
institutions possess no advantages over the others, as none 
of them have the power of conferring degrees. 

The course of study in this class of colleges usually 
comprises instruction in book-keeping, with its application 
to the various branches of trade, manufactures, etc.; mer- 
cantile arithmetic, penmanship and business correspond- 
ence ; lectures on the usages of trade, negotiation of 
business paper and the most useful branches of commer- 
cial law. 

When conducted with ability and integrity, commercial 
educational establishments rank among the most useful 
institutions of learning of the day. Though of compara- 






THE MERCANTILE PROFESSION. 97 

tively recent origin, they are to be found in most of the 
larger cities of the Union, and receive a liberal patronage 
by all classes of the community. Professional men, me- 
chanics, farmers, and, in large cities, ladies, are to be 
found among the number who consider their education 
unfinished until they have passed a commercial course in 
one of these, and merited a diploma. 

It may be safely asserted that in no institution of learn- 
ing is there as much useful information obtained in so 
short a time, and at such trifling expense, as in commercial 
colleges. 

Business men may obtain a knowledge of book-keeping, 
as applied to their own business, in .a few weeks, while 
most youths might be profitably engaged in a college at 
least one year. 

PERSONS EMPLOYED IN BUSINESS HOUSES. 

The persons employed in mercantile houses are: Book- 
keepers, Correspondents, Solicitors, Salesmen, Travelers; 
Entry, Bill, Shipping and Engrossing Clerks, Junior 
Clerks or Boys, Porters, Coopers and Draymen. 

DUTIES OF THE VARIOUS OFFICES. 

Book-keeper. The book-keeper's place of business is in 
the counting-room. His duty is to keep the accounts of 
the establishment, in a variety of books for that purpose, 
to receive and pay out all moneys, and deposit money in 
the banks for safe keeping, to make out bills, render ac- 
counts, statements, etc., from the ledger and sometimes to 
conduct the business correspondence. 

Correspondent. The business of the correspondent is to 
reply to letters of inquiry, and to write all letters of bus- 
iness connected with the house, etc. In extensive im- 
9 



98 NELSON'S COMMON-SCHOOL ARITHMETIC. 

porting houses, the correspondent is usually a person who 
understands some three or four of the modern languages. 

The Second Boole-keeper assists the first book-keeper. 
He copies or transfers to a journal or day-book the items 
found in the sales-book, journalizes and posts to the 
ledger. It devolves upon him to make out bills, accounts, 
and to assist in the counting-room generally. 

The Solicitor traveling agent belongs to the broker's 
class. His business is to solicit orders and secure buyers 
for houses with which he has made previous arrangements. 
Accordingly he is found in the hotels, on the steamboats 
or at the railroad depots. When he finds a buyer, he con- 
ducts him to the store for which he is operating, and, if 
not under salary, receives a commission for his services. 

Clerk is a general term applied to all employes, the 
porter, drayman and cooper excepted. 

Salesman. The duties of the salesman consist in un- 
packing, marking and arranging goods for sale, receiving 
customers and selling to them. 

In some houses, the salesman receives a commission for 
the amount of trade he influences, and in all places the 
amount of his salary very much depends upon this cir- 
cumstance. 

These statements apply principally to wholesale busi- 
ness. 

The Skipping Clerk receives and examines goods to see 
if they agree with the conditions of the bill of lading, and 
attends to the shipping of goods from the establishment. 
These he enters in a book for that purpose, called the 
shipping-book. 

The Entry Clerk records the sales made by the sales- 
man in a book called a blotter or sales-book. 

The Bill Clerk makes out the bills or outward invoices 
from the sales- book. 



THE MERCANTILE PROFESSION. 99 

Entry and bill clerks should be rapid penmen and ex- 
pert in figures, if they would command liberal salaries. 

The Engrossing Cleric assists generally, sometimes in 
the counting-room, but more generally as entry or bill 
clerk. He is simply a copyist.* 

Junior Clerks are usually boys from 12 to 17 years of 
age, whose duty it is to run on errands, pack and unpack 
goods, mark boxes and packages, and keep sale-rooms in, 
order. 

After acting in the capacity of junior clerk for two or 
three years, they are promoted to more lucrative and re- 
sponsible offices. 

The youth who would aspire to a high degree of use- 

fulness~in his profession, should not rest content with the 

I / 1 

opportunities for improvement afforded by the store or * j J 

sales-room. His evenings should be devoted to useful 
study, until he acquires proficiency in all the various 
branches of mercantile business. 

The study of Freedley's excellent " Treatise on Busi- 
ness" is highly recommended, especially chap, iii, page 46. 

Heading-rooms, libraries, mercantile associations, mer- 
cantile and commercial colleges are to be found in most 
of the larger cities of the Union, and should form the 
places of resort for young men of this class in preference 
to questionable places of amusement, too much frequented 
by them. 

Porter. The business of the porter is to open and close 
the store, keep the store and counting-rooms clean and in 
order, pack and unpack goods, assist in handling and 
weighing heavy goods, marking packages, etc. 



~*In many houses, the whole business of clerking is performed by 
one person, while in others many more offices are called into requi- 
sition than are noticed here. 



r '// 

m 



100 NELSON'S COMMON-SCHOOL ARITHMETIC. 

The office of porter is a more responsible one than most 
people imagine. By the faithful discharge of its duties, 
hundreds of men in this country have been placed in pos- 
session of respectable retail houses. 

In some large establishments, there are two or more 
porters engaged, between whom the duties of the above 
are divided. 

Cooper. In liquor and heavy sugar establishments, pack- 
ing-houses, etc., the services of a cooper are required, 
whose duty it is to open and close hogsheads, barrels, etc., 
and to repair damages to which such articles are subject 
in carrying. 

Drayman. The drayman acts as "carrier" between the 
etore and depot, landing or wharf. He usually keeps a 
book called a dray-book, in which are entered the con- 
tents of each load. This is signed by the clerk at the 
place of delivery, and when the entire shipment is made 
the amount is entered in the Bill of Lading. 

XL BILLS INVOICES. 

97. WHEN goods are sold, it is the duty of the mer- 
chant, or one of his clerks, to make out a statement of ] 
the quantity, kind and price of each article, for the sat- \ 
isfaction of the purchaser, and to enter at the foot of such > 
statement the whole amount of the purchase, with the ; 
payment received, if any, or the terms of settlement. If 
the goods are bought to sell again, this statement is com- 
monly called an Invoice; otherwise it is called a Bill, es- 
pecially by the purchaser. 

A bill or invoice is sometimes delivered to the buyer at; 
the time of purchase; but it is usually sent with the 
goods, or, if the buyer resides at a distance, by mail. 

An invoice should specify the place and date of sale, 



THE MERCANTILE PROFESSION. 101 

the name of buyer and seller, a description of the goods, 
the price of boxes, etc., used for packing, charges for in- 
surance, and, when 'payment is not made, the terms of sale. 
When goods are received, the quality and quantity are 
compared with the invoice, and the selling prices made 
out from it, after which it is filed away or pasted in a 
book for future reference. 

98. It is the custom of merchants to have their bill 
heads (heading of bills) printed, with names of city and 
street, number of house, and such other matter as will fa- 
cilitate the labor of clerks, or otherwise advance the inter- 
ests of the business. A specimen form of such heading 
will be given in the bills that follow. 

99. Filing Bills. When there are many bills on hand 
designed for collection, they should be folded neatly of the 
same length and breadth, and have the names, addresses 
and amounts written on the outside at the top. A gum 
band will then keep them firmly in their places, and per- 
mit their being delivered without the trouble of opening 
for examination. 

100. Retail bills are rendered periodically, by the 
month, quarter, half-year or year, according to agreement 
or the usage of the house. When a pass-book is not 
kept, it is well to have a memorandum of each purchase, 
so that in rendering his final bill the merchant need not 
insert the items. 

101. Account or Statement. The final bill of a mer- 
! chant now goes by the name of account or statement. The 
! head contains the date upon which it is drawn, and the 
! word "To" substituted for "Bought of." In the margin, 
I on the left, are the dates of the several purchases, with tho 
| words, "For amt. rendered," or "Amt. pr. bill rendered." 

102. In making out bills, the three requisites are ra- 
j pidity, legibility and accuracy. The principal is accuracy. 



102 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



In business, it is not enough to be right after one or two, 
or perhaps repeated, attempts. The clerk should be cor- 
rect the first attempt, and generally is so. Boys designed 
for business pursuits ought to spend much time at bill- 
making, until they acquire familiarity with the numerous 
abbreviations, and can make out a bill from dictation al- 
almost as rapidly as the items can be called off. 

103. Finding the cost of a number of articles at a cer- 
tain price, and placing the amount opposite, is called, in 
bill-making, extending; adding the columns, footing up. 

104. Receipt on a Bill. A clerk or agent may write 
the name of his employer to a receipt and it will be good, 
if he write his own initials or last name underneath. 

BOOKSELLERS AND STATIONERS' BILLS. 

CINCINNATI, June 16, 1866. 

MR. HORATIO NELSON: 

Bought of R. W. CARROLL & CO., 

PUBLISHERS, BOOKSELLERS & STATIONERS, WHOLESALE & RETAIL, 
117 WEST FOURTH STREET. 

TERMS : In making orders, be particular to avoid mistakes. 

All claims for Errors or Damages to be made within five days of receipt of goods 



1 Gro. Pen-holders, 




2 


35 




*[ u u u 






75 




3 Doz. Paper-Cutters, .... 
2 " Ebony Rulers, .... 
1 Rm. Bill Cap, 


2.00 
3.75 


6 

7 

7 


00 
50 
00 




2 Letter Cap, No. 1400, . . 
1 2-Qr. Blank Book, .... 
3 Qr. Blotting Paper, .... 


8.25 
40 
2.25 


16 

6 


50 

80 

75 


$47.55 



Rec'd Paym't. 



R. W. CARROLL & Co. 
DAVIS. 



Gro., Gross; Doz., Dozen; Qr., Quire; Rm., Ream. 



4 



BILLSINVOICES. 103 

BOOKS STATIONERY. 

105. Books, stationery, etc., at wholesale, are usually 
sold by the dozen. Paper is sold by the ream, bundle or 
pound. Writing paper is put up in half-reams; printing 
paper in bundles of two reams. 

In the exercises which follow, the teacher will find it to 
the advantage of scholars to have them write from dicta- 
tion. The bills may be made out in favor of the learner 
or otherwise. 

3. J dz. Hooker's Nat. Philos., $14.40; 1 rm. cap. $6.50; 
1 do. bill, $7.00; 2 dz.!2-in. ebony rulers, $3.75; 2 dz. paper- 
cutters, $2.00; 1 bx. crayons, 35 c; 12 dz. pass-books, 40 c 

4. 2 dz. mucilage, $2.75; 1 dz. carmine, $2.00; 2 dz. 
tin cutters, $2.00; 2 dz. rulers, $4.50; -J- rm. natl. cap, 
$5.50; 3 dz. No. 4 pass-bks 44 c rm. treas'y cap, $8.40; 
1 qr. blotting, $2.25. 

5. 1 bx. 4560 5^ envelopes, $2.25; 1 do. 8J Manilla, 
$2.75; 1 dz. Lincoln portraits, $3.00; 2 dz. check-books, 
$12.24; 1000 penman's blanks, $20.50; 21 T 9 3 dz. bill- 
books, $2.16; 4 dz. do. 2 qr. ea., $2.50; 20 dz. check, 
$2.16; 20 dz. inv., $1.00; 20 dz. day-books, $1.80; 60 
dz. journals, $1.80; 20 dz. ledgers, $1.80; sunds,, $7.00. 
Credit cash, $210. 

6. 1 copy'g press, $10.00; | dz. Rec. of a Country Par- 
son, $20; T V dz. Wordsworth's Poems, $25.00; T \ dz. Be- 
ginning Life, $12.50; y 5 ^ dz. Heads and Hands in the 
Wrld of Labor, $12.00; i Essay on Woman's Work, 
12.50; 2 T % do. Nat'n'l Lyrics, $4.80. 

Answers: $55.80, $30.21, $33.75, $31.02, $150.16. 

SHOE BUSINESS. 

106. Shoes are usually sold at wholesale by the dozen, 
boxes furnished gratuitously. 



104 NELSON'S COMMON-SCHOOL ARITHMETIC. 

Pairs. 

7. 12 Wos. Goat Tips Bals 1.45 

6 Miss " " " 1.75 

12 Childs " " 1.05 

12 Calf " 1.05 

12 Wos. Goat Tips " 2.50 

10 " Last. Lace Cong 2.00 

6 " But. Gait 2.90 

6 Miss Kid Cong. " 1.90_ 

Numbers 
on boxes. Pairs. 

8. S 12 Child Serge Lace Heel'd Gaiters. 65 

474 13 Ladies Kid ch. nl. Balmorals 1.30 

533 12 " Goat D. S. " 1.28 

2386 11 Wos. Kid Cong. D. S. Boots.... 1.374 

2449 12 " Serge Cong. Gaiters 1.20 

2593 12 " " " " 1.124 

2475 24 Child Gai. Peg-Heel'd J. L. Boots 45 

2586 12 " End. u " cop. tip. 45 

2580 24 " Buff " Lace " " 40 

2575 12 Miss GaL " J. L 674 

2413 15 " Goat " Cong Gait... 95 

2517 12 " " " " . . . 60 

2591 9 Ladies Goat, tip. ch. nl. Bals 2.00 

2431 12 Youths Buff Peg-H'1'd " 724 

2461 12 Ladies Calf " tip Bals.. 1.25 

2588 12 Kid ch. H'l'd . . 1.50 

2589 12 Peb. Calf ch. HTd tip Bals 1.50 

2590 12 Miss Kid ScT. Welt " 1.05 



ABBREVIATIONS. TFbs., Womans; D. 5., Double soled ; Cong., Con- 
gress; Grd.j Grained; End., Enameled; J. L., Jenny Lind; Cop* 
Tip., Copper tipped; Ch. NL, Channel nailed; Hid., Heeled; Sd, 
Wit., Sewed Welt; Peb. Cf., Pebbled calf; Bals., Balmorals; But. 
Oait., Button Gaiters. 



BILLS INVOICES. 105 

MILLINERY. 
9. 60 12 15-Braid Bonnets @ J0.62J 






68 6 




it 


1 


.25 


70 4 


7-Braid 


a 


u 


1 


.50 


. 80 2 


7 " 




a 


u 


3 


.00 


86 2 


7 " 




n 


u 


3 


.75 


6 


PCS. 


No. 1 Tafft. Ribbon 


a 




15 


5 


a 


" 2 


u u 


a 




28 


3 


a 


" 4 


u u 


u 




48 


2 


u 


" 6 


it it 


u 




75 


1 


u 


" 12 


u u 


tt 


1 


.10 


3 


tt 


Bonnet Ribbon 


u 


2 


.00 


2 


a 


u 


u 


a 


2 


.50 


3 1 


Box 


Ruches. . 




a 


1 


.50 


415 1 


ti 







u 


2 


.25 


210 i 


Doz. 


Bunches 


Flowers 


u 


18 


.00 



J " Feathers " 36.00 

1 PC. Black Silk, 20 yds " 87J 

GROCERY BUSINESS. 

107. TARES. In Cincinnati: New Orleans sugar, in 
hogsheads, 10%; Cuba and Porto Rico, 12%; sugar in 
boxes, 15%. Rice in tierces, 10%; indigo in ceroons, 
11%; in boxes, actual tare; salt in barrels, each 30 Ibs.; 
coflee, cotton, spices, feathers and salt, in bags and bales, 
no tare; manufactured tobacco in kegs and boxes, (not 
enumerated,) actual tare; madder in casks, actual tare; 
lard and bacon in packages, actual tare. Lard kegs tared 
after being emptied.* 

* Butter in firkins is subject to an allowance of 2 Ibs. for soakage; 
in rolls packed in half barrels, 1 Ib.j barrels, 2 Ibs.; tubs 50 to 
70 Ibs. 1 Ib. 



106 NELSON'S COMMON-SCHOOL ARITHMETIC. 

In New York: The tares differ slightly from the above; 
"but the list being a long one, it is reserved for the revised 
edition of " Nelson's Mercantile Arithmetic." 

Extra Charges may be made for drayage, insurance, 
cooperage, storage, boxes, bags, etc., in which goods are 
packed in the store. No extra charge is made for the 
original package. In the bills which follow, some of these 
charges are introduced. 

10. 1 Hhd. N. 0. Sugar, *^oo **1080 Ibs. .@$.07 
4 Brls. N. 0. Molasses/H 169 gals ---- " 35 

1 Trs. Eice, g 630 Ibs ............. " 4 

20 Bags Rio Coffee, 3200 Ibs .......... " 11 

2 Half Chests Black Tea,** o 1 ^** 72 Ibs. " 25 

1 0028 

3 " " Yng. Hyson do. 150 Ibs. . " 50 

1 " " Imperial do. 60 Ibs.." 40 

2 " " Gunpowder do. 110 Ibs.. " 60 
1 " " Oolong Blk. Tea, 45 Ibs. " 40 



VJ 

1 


Box 


tiAWAUVt. V^ A U.J-iC*il.iJ. \_fJLi 

5 lump Tobacco, 


l l\ 108 Ibs.. 


cc 




25 


1 


a 


pound lump " 


'4J 124 Ibs.. 


(C 




20 


1 


u 


Va. pound " 


4o 120 Ibs.. 


a 




35 


1 


a 


8 lump " 


J |g 125 Ibs.. 


u 




22 


20 


Brls 


Rect. Whisky 


800 gals 


a 




17 


4 


a 


Ginger Wine, 


160 gals 


a 




60 


\ 


Cask 


French Brandy, 


40 gals 


a 


4 


.00 


% 


a 


Port Wine, 


45 gals .... 


a 


2 


00 


10 


Brls. 


Bourbon Whisky, 


405 gals .... 


a 


1 


.00 


J 


Brl. 


Holland Gin 


20 gals.... 


u 


1 


.50 


5 


% * 


F , 

























* Gross Weight. J Gallons in each barrel. 

{Tare, or weight of bag, box, etc. **Net Weight. 



BILLS-INVOICES. 107 

The small figures on the left indicate the prices of boxes, 
barrels, etc. 

11. 100 Boxes Cheese, 4 '?<> 3690 @$ .08 

30 Firkins Butter, 3 f<j<> 2820 " 15 

100 Boxes $2 Starch, 4810 " 5 

100 " *** Star Candles, 4000 " 20 

20 Bbls. $ * 5 Lard Oil, 810 gals " 85 

50 " Mess Pork "16.00 

10 Tierces S. C. C. Hams, 3 |f g 3000. " 11 

30 Kegs Lard, 'J'g 1334 " 12 J 

15 Bbls. Mess Beef 15.00 

Com. for purchasing, $1521.75 " Z\% 

Drayages 16.00 

Insurance on $5000 59.88 

12. 1 bag Pepper, 103 10J 

1 " Allspice, 128.. 10 

4 dz. Shakers' Brooms 2.40 

1 Em. Cap Paper 4.00 W% off. 

1 " Med " 6.00 " " 

1 " D. " 8.00 " " 

5 bxs. 10 Ger. Ex. Soap, 297. 7 

1 Keg Soda, 112 Ibs 5J 

1 bx. 20 Saleratus, 61 Ibs.... 5J 
1 " 20 Saltpeter, 47 u .... 9J 

Drayage 79 

13. 10 Bbls, -.<> Sugar, 

246 23 245 20 

233 18 246 17 

250 21 275 21 

227 22 232 19 

239 21 266 _25 

2459 207=2252 lbs.@12|c, $ 
Drayage 1.60 



108 NELSON'S COMMON-SCHOOL ARITHMETIC. 

14. 1 Bbl. ' 35 Lard Oil, 41 85 

1 Hf. bbl. 75 98% Alcohol, 21 54 

1 Dz. Washboards 2.25 

1 ' Sk. Bar Lead, 96 Ibs 6J 

1 Pkg. Yarn, 28 Ibs 26 

2 Doz. No. 2 Brooms . . . , 2.40 

1 Trc. Ex. S. C. Drd. Beef 2 25 

1 Ke<r lOd Nails 3.00 

| Bbl. 5 Lard Oil, 23 95 

Drayage 75 

15. 2 Cases H. Cheese, 46 Ibs 28 

4 " S. W. R. Cheese, ! ?| 112 Ibs.. 22 

3 " E. D. " 87 78 u -fl 18 

2 " N. W. R. " 2 5 21 " .. 17 

1 Bbl. Cuba sugar, 2 |f 202 Ibs 13 J 

4 Boxes MR. Raisins 4.40 

2 " " " 2.40 

2 i " " .. 1.45 

3 Jars Prunes 45 

5 Drums Figs, 37^ Ibs 24 

1 Case 100 half boxes Sardines 42 

1 Box American Castile Soap, 28 Ibs.. 16* 

1 " French " " 35 " .. 16 

1 Doz. Cox Sparkling Gelatine 6.00 

2 Boxes * Saleratus, 128 Ibs 7^ 

5 Half boxes Star Candles, 100 Ibs 24 

1 Bbl. 25 Com. Smok. Tobacco, 100 Ibs 12 

1 " 2S S. F. " " 83 " 25 
2 Boxes 40 Starch, 96 Ibs 7 

2 " Soda, 92 13 

Sk., sack ; Pkg., package ; W. R., Western Reserve. 2 boxes 
raisins, 2 half boxes. 




BILLS INVOICES. 109 

16. 1 Bbl. 25 Soft Refined Sugar, 240 Ibs. . @ 16 
1 25 Hard " 220 .. " 17 
1 " 25 Granulated " 212 " .. " 25 

1 Sack Java Coffee, 136 Ibs " 41 

1 " Rio " 156 " " 28 

1 " Laguayra, 111 Ibs " 30 

| Chest Y. H. Tea, 8214^68 Ibs... " 1.25 
I " Black 53 12=41 Ibs... " 1.35 
J " Y. H. Canton Tea, f * 68 Ibs. . . " 90 

i " " Fine, ?o 66 Ibs "1.60 

-J " Oolong Common Tea, }$ 41 Ibs. " 90 
1 Sack African Peanuts, 2 bushels .... 2.40 

1 " Roasted do., 3 bushels 3.10 

1 Bbl. 25 Dried Peaches, } J 95 Ibs. . . 21 
1 " 25 " Apples, 105 85 Ibs.... " 15 

1 Box 20 German Soap, 64 Ibs . .. " 12 

1 " 2 Common " 64 Ibs " 12 

1 " 20 Cincinnati Extra do., 61 Ibs. . " 14 

1 Bbl. * 5 Prm. Wh. Beans ~?j> 216 Ibs. " 2-f 

2 Boxes 2 doz. Baking Powder " 4.80 

17. 2 Sks. Rio Coffee, 323 25 

1 Bbl. Molasses, 45 J 60 

1 " 37 Ricej 23519 10 

2 " 75 Sugar, iJl=?g 12 

Drayage 50 

DISCOUNT. An abatement entitled discount is often made on the 
bill for cash or when goods have fallen in price. When making 
such abatements, the clerk should remember that he is discounting 
the profits as well as the first cost. For instance, I buy goods at 
$100, and sell them at a profit of 50 per cent., which makes the 
price $150. Now, if this is discounted at 40 per cent., it does not 
follow that a gain of 10 per cent, is made. 40 per cent, of $150= 
$60, which, taken from the selling price, leaves $90, making, in- 
stead of a gain of 10 per cent., a loss to that amount. 



110 NELSON'S COMMON-SCHOOL ARITHMETIC. 

18. 20 Bbls. Molasses, 

43J 44 43 

42J 



44J 


411 


44^ 




43* 


44 


441 




44 


42^ 


42J 




43 


43J 


411 




44 


43 










865J *2@75o 


2.00 











19. 10 Tcs. Shaffer's Hams 3 *'<> $0.21 

5 " A. Spence's S. C. do. ^Jf 25 

1 Bbl. 25 20 dz. Tongues 1.00 

Com. on $1013.78 2%% 

Exch \% 

Drayage 2.00 

Answers: $86.24, $262.26, $1046,23, $138.73, $285.60, 
$185.51, $4152.87, $1665.97, $649.63, $688./5, $131.90, 
$228.74, $130, $97.09. 

NOTE. When merchants can not fill the orders from their store, 
they sometimes buy elsewhere, and charge their customer cost and 
a commission, as in the last bill. 

The Teacher should not confine himself to the few exercises in 
bill-making given in this work, as he will find it both interesting 
and profitable to the learner to represent other kinds of business in 
the same way. Any number of bills can be dictated from price 
lists or prices current, to be had in any large city. 
* Leakage. 



"^ , 



BILLS INVOICES. HI 



DRY GOODS * 

Yds. Price 

20. 1 PC. H. A. g- Blea. Muslin 45 25 

1 " A | Light " 43 15 

1 " D. Q. | Fine Bro. Muslin. . . . 43 22 

2 " P. I Muslin 70 2 24 

1 " L. Check 68 1 26 

1 " W.Jeans 46 2 30 

A3" Fey Prints 126 1 15 

B 1 " " " 37 2 16 

C 9 " " " 355 2 17 

1 " BuffChambray 20 2 35 

1 " Dom Gingham... 35 25 

1 " L " 60 2 25 

2 Doz. Coats' Spools 1.10 

1 " | Hose 3.00 

1 " Ladies' L. W. Hose 6.75 

No.4 1 Pk. Pins 70 

31" "..... 75 

21" " 90 

1 Doz. Lin. Hdkfs 6.00 

1 " H. S. " 9.50 

f 1 PC. Sheetgs to fill 35 2 24 

2 Bxs 5 and Strapping 5 2.00 

* Though practically correct, some of the answers to these bills 
will be found mathematically wrong. Accuracy in cents has been 
sacrificed in conformity with business usage, which often rejects 
fractions in extensions, alternately adding a cent and rejecting a 
half or fourth. The letters "H. A." etc., indicate the^rade of nius- 
liiis, or are the initials of the maker or factory ; the figures and 
letters in the margin are marked in the wholesale house to distin- 
guish lots of nearly the same kind from one another. 

tNot in the order; but will not be objected to by the buyer, as it 
prevents goods from shifting in the case. 



112 NELSON'S COMMON-SCHOOL ARITHMETIC. 



21. 4 


PCS, 


, Prints 


. 185 3 




17 




1 


tt 


Jeans 


. 45 1 




30 




1 


a 


Cottonade 


. 38 1 




37 s 




5 


it 


Chamb. Gingham 


. 154 2 




35 




2 


it 


Luster 


. 101 2 




30 




2 


Doz 


. Skirt Braid. 




1 


.00 




10 


u 


Coats' Spools 




1 


.10 




No. 101 


u 


Miss's Hoops 








5.00 


181 


M 


a tt 








9.00 


201 


u 


n it 








10.00 


30 4 


a 


2 Gore Miss's Hoops 


2 


16 


.00 




22. 1 


PC. 


Fancy Cassimere 


27f 


2 


.20 




1 


u 


French " 


28| 


2 


.85 




1 


a 


Mixed " 


31 


2 


.90 




1 


M 


Amoskeag A Ticks 


GO 




47 




1 


U 


Glassgow Gingham 


40J 




27 




1 


u 


Victoria " 


37 




42 




1 


a 


Shawmut Can. Flannel .... 


35J 




34 2 




3 


u 


LX.L.Wl.do.41,64,34.. 


139 




73 




1 


u 


Roan. Striped Shirting .... 


37J 




31 




2 


a 


Fey Slk Dress Goods, jj.. 


71 


1 


.10 




1 


tt 


Village Green Checks 


37| 




23 




4 


tt 


Lonsdale Checks f? ?<) 


143 




21 




1 


it 


Blk French Broadcloth . . . 


29} 


4. 


.25 




1 


tt 


English " 


37J 


5 


.00 




2 


tt 


Bolivar Denims \ g 


98 




20* 




1 





Charter Oak Denims 


60 




22 




1 


a 




38| 




17 a 




2 


IK 


Duchess Delaines, 31 1 36 1 . . 


67* 




30 




4 


a 


Semper Idem, 40 38 27 1 36. 


1151- 




32 2 




2 


a 


Barre Brown Sheeting \ g . . 


76 




34 




1 


tt 


Great Falls do. do.. 


42J 




28 
















2.50 



BILLS IN VOICES. 113 

23. 75 48 Doz. Gent's Shakspeare Coll 2.00 



6 60 


u 




Ladies Plea " , 




1.26 


1 48 


N 




" BvronEmb. " , 




1.50 


1 45 


U 




" Gar. " " , 




1.50 


1 48 


it 




u p er a tt 




1.50 


1 120 


tt 




" Sq. Gar. C.&C.E. 


Coll. 


1.00 


1 29 


a 




" Mull. Edge Gar, 


u 


1.50 


1 20 


tt 




Vic. Cuffs, 2 C ., 




2.50 


1 2 


a 




" ..Octagon C.&C.E 


.Sets 


4.00 


1 5 


N 




" Sqr. Emb. 


u 


5.50 


1 8 


(C 


" Point Wht. Trim. " 


4.50 


16 47 


n 




Shirt Fronts 




3.50 


24. 11 10 PCS. 


4 
4 


Shirting Linen 


294 


$0.37 


12 10 


a 


4 
4 


tt it 


297 


40* 


13 10 


tt 


4 


it 


300 


44 


14 15 


(1 


4 


u 


433 


49 


15 15 


u 


A 


u u 


447 


55 


16 10 


tt 


1 


U It 


285 


60 


17 10 





4 


tt U 


287 


85 


25. 1342 2 


Doz. 


Napkins 




2.75 


1343 2 


u 




u 




3.40 


1356 2 


a 








4.50 


1270 2 


u 




it tl 




4.00 


1138 4 


Felt 


Table Covers 




2.85 


1416 1 


PC 




fl6 2 


57* 


1417 1 


tt 




" " "3 


26 


62 2 


1418 1 


a 




" u " 4. . 


26 


70 


1419 1 


n 




tt u a /) 


27 


75 


1421 1 


u 




u a ^ 


26 


87 2 


1422 1 


tt 




a u a g 


27 


1.00 





Emb., Embroidered; Gar., Garote; Per., Persiguey; C. $ 2 <?., 
Cord and 2 Cord; Vic., Victoria. 
10 



114 NELSON'S COMMON-SCHOOL ARITHMETIC. 
QUEENSWARE. 

MV 

3 Doz. Edged Plates $0.40 

10 " " " 50 

5 " *CC " 50 

" " Dishes, ea, $1.75, 2.25, 2.75. 
1 " " Bakers, ea. $1.50,2.00, 2.50. 

1 " " Beaded Nappies, ea. 1.75, 2.25 

| " " Tureens 3.50 

2 " " Bowls 80 

2J " " " 60 

3 " " " 50 

\ " " Pitchers 3.50 

1 " Colored " ea. $2.50, $4.00 

4 " " Bowls 87 2 

5 " " " 65 

6 " " " 55 

9 Sets CC Teas 20 

36 " Painted Teas 20 

Crate 1.00 

27 HAT BUSINESS. 

\ Doz. Men's Black Cass. Hungarian 21.00 

\ " " " " 27.00 

6J " " " " 33.00 

| " " " Broad Brim Wool. .. 14.00 

1 " " Wove Senate 12.00 

J " " " Cashmerette 15.00 

\ " Boys' " Hungarian 7.00 

4 " " Caps, assorted 12.00 

i " " " " 9.75 

J " " Cloth Caps 9.00 

\ " Children's Fancy Caps 8.00 



* CC, cream colored. 



i 






BILLS INVOICES. 115 



28 HARDWARE. 

25 Bars 1 JX J Bar Iron ......... 750 Ibs. $0.03 



6 Bund. Round " ......... 625 " 4 

4 11 x| Dandy Tire Iron.. 500 " 3 s 

3 " 1 X| Horseshoe Iron^ 

4 " 1 X T 7 e " " [ 975 " 4 
2 " 1 X -\ " "3 

2 Sets 1JX4 L. Pitts Springs^ 

3 1JX5 " " " t- 500 10 

2 " 2 v5 u " u 

1 " 2 X8 ". " " 

4 Slabs 14 X | Steel ........... .600 " 7 2 

3 Doz. Amos No. 2 Shovels ............ 10.50 

2 " Rowland's No. 2 Shovels ......... 8.50 

1 " No. 2 Spades ......... 8.50 

6 " No. 5684 Pocket Knives ......... 7.50 

3 " " 4215 " " ....... 4.50 



Answers: $238.88, $160.39, $56.73, $194.92, $1234.04 
$832.60, $232.53, $981.51, $318.25, $735.63. 

MERCHANT TAILOR'S BILL. 

9 - NEW YORK, Ap. 5, 1867. 

MR. A., 

To B. 

Jan. 3. For 1 Blue Beav. Overcoat 70.00 

Feb. 9 1 Pr. Blk Doeskn Pants 23.50 

$93.50 
Mar. 2. Cr. By Cash 50.00 

$43.50 

To THE TEACHER. The learner ought to be taught that bills are 
seldom punctuated, and that even the dollar or cent signs or @ are 
rarely used. It is proper, however, to insert the dollar sign at foot- 
ings. 



116 NELSON'S COMMON-SCHOOL ARITHMETIC. 

FARMER'S BILL WITH CREDITS. 

RICHMOND, Nov. 1, 1869. 
MESS. TRADER & Co., 

To JONATHAN FARMER. 

Jan. 3. For 3 Durham Heifers 25.00 75.00 

" " 1 2-yr. old Bay Colt 50.00 

Ap. 11. " 140 2 Bu. Corn 50 70.25 

" " 5 2 Dz. Chickens * 1.37 2 7.56 1 

9. " 123 2 Ibs. Butter 18 3 23.15* 

Cr. "$225^97 

Ap. 4. By 25 Hogs, 3147 Ibs 3 2 110.14 2 

9. 1 Plow 7.62 2 

" Cash 108.20 $225.97 

JONATHAN FARMER. 

MECHANIC'S BILL. 

ST. Louis, July 3, 1867. 
MESS. BUREAU & Co., 

To JOHN WORKMAN. 

Jan. 1. For Cab't W'k on 3 dz. hf-Fr. Bed's. 20.00 60.00 
Feb. 9. " " " " 4" 2-pn'ldo... 30.00 120.00 

Cr. 180.00 

Jan. 1. By Cash on Ace 10.00 

" 8. " .Order on W. Shoemaker f 15.00 

" 15. " Cash 20.00 45.00 

Settled by note at 90 ds $135.00 

JOHN WORKMAN. 

When the signature of the merchant is required to a part pay- 
ment, the phrase Balance due should be written opposite the amount 
unpaid. 

* Express the 137 2 in mills before multiplying. 
t For form of order and note see index. 
Hf. Fr. Bd's.> half French bedsteads; pnL, panel. 



\ 



REDUCTION. 117 

DRESS-MAKER'S BILL. 

BOSTON, Sep. 9, 1866. 
MRS. AFFLUENT, 

To M. E. FASHION. 

For Making 1 Moire Antique Dress 25.00 

" Trimming do 75.00 

$100.00 



XII. COMPOUND NUMBERS. 

BESIDES the distinction made between numbers in chap- 
ter viii, they may be divided into simple, and compound. 

108. A simple number may be abstract or concrete, of 
one denomination, as 27 men, 35 dollars. 

109. A compound number is always concrete and com- 
posed of more than one denomination, as 157 dollars 50 
cents, 29 pounds 14 shillings and 6 pence two numbers, 
each expressing one sum of money. 

110. Reduction of compound numbers. 

111. REDUCTION is the process of changing concrete 
numbers of one denomination to those of equal value in 
another. If I multiply 5 bushels by the number of pecks 
in a bushel, I reduce them to pecks, and thus change both 
the number and denomination, while I preserve the value.* 

112. Changing the denomination from a higher to a 
lower, as bushels to pecks, is called reduction descending; 
while the reverse process, as changing pecks to bushels, 
is called reduction ascending. 

A few examples will suffice to teach all that is neces- 
sary to be known on this subject. 

* On page 70 are several exercises which properly belong to this 
subject. 



118 NELSON'S COMMON-SCHOOL ARITHMETIC. 

1. To reduce 35 acres to, or represent them, in square feet. 

35 Acres EXPLANATION. In 1 acre there are 4 roods, 

4 in 35 acres there are 35 times 4 roods (35X4) 

TT7 r> 7 or 140 roods; in 1 rood there are 40 square 
rods, in 140 roods there are 140X40 or 5600 
sq. rods ; in 1 sq. rod there are 30J sq. yards, 



5600 Sq. Rods j n 5^00 S q. rods there are 5600X30J or 169400 
sq. yards ; in 1 sq. yard there are 9 sq. feet, 



30j 



168000 in 169400 sq. /ards there are 169400X9 or 

1400 1524600 sq. feet. 

REMARK. It will be observed that the 
game regulfc would haye been produced by 

multiplying the 35 by 43560, the number of 



1524600 feet in an acre. 

When there are items between the highest and the low- 
est denominations, they should be added, as shown in the 
following example: 

2. In 75, 13s, 6Jd how many farthings? 

75 13 6J 
20 

1500 Shillings in 75. 
13 Shillings added. 

1513 Whole number of shillings. 
12 



18156 Pence in 1513 shillings. 
6 Pence added. 

18162 Sum of pence. 

4 Par things in 1 penny. 

72648 

9. 



72650 Parthings in 18162J pence. 

To add the items mentally, the 3 shillings would occupy the place 
of the first 0; then, multiplying the 5 by the 2, we would obtain 10 
to which might be added the 1, making 11; then, multiplying the 
7 by 2, we get 14 and the one carried, making 15 or 1513 at once. 



REDUCTION. H9 

3. In 3 miles 21 rods how many yards? 

4. In 145 tons 25 Ibs. of hemp how many pounds? 

5. How many pence in 197, 17s, 9d? 

6. How many farthings in 57, 13s, 6? 

7. In 93 barrels of apples how many pecks, each barrel 
containing 2 bushels 3 pecks? 

Answers: 324825, 55370, 47493, 1023, 5395. 

113. To reduce concrete numbers of a lower denomina- 
tion to those of a higher, the process will be the reverse 
of the last. 

8. Reduce 72650 farthings to pounds. 

4)72650 Farthings. In farthings there will be one fourth 
1 9M m t\9 9 P as man y P en ce; in pence one twelfth as 

~ 1 ' many shillings; and in shillings one 

2 1 0)151 1 3 6 Skill. twentieth as many pounds. The re- 
75 13 6 1 raainders are 13 shillings, 6 pence and 2 

farthings or J penny. 

9. Reduce 4163 linear inches to yards. 

10. Express 31456739 minutes in years, months and 
days, allowing 365 days 6 hours to the year.* 

11. Reduce 456372 farthings to pounds. 

12. At 1 mile in 4J minutes, how many miles would a 
locomotive run in 5 hours? 

13. A ship sailed 3000 miles in 16 days; what was her 
average speed per hour? 

14. In 35 cubic yards how many cubic inches ? 

15. In 5J square rods how many square feet? 

16. Divide a log 55 feet in length into 15 equal -parts, 
and express the result in feet and inches. 



* Reduce the year to minutes and divide them into the minutes 
in question, which will give the number of years. The remainder 
being minutes, may be reduced to hours, etc., as in reduction. 



120 NELSON'S COMMON-SCHOOL ARITHMETIC. 

17. What will be the cost of 35f bushels of strawber- 
ries at 15 cents a quart? 

18. At 5 cents for 3 sheets of paper, how much money 
can be obtained for 35| reams, allowing half price for the 
outside half quires (35-J- qrs.) of each ream? 

19. A merchant buys 15 barrels of potatoes, containing 
2 bushels 2f pecks each, at $1.25 per bushel, and sells 
them at 25 cents a half peck; how much does he make, 
allowing \ a peck per barrel for loss in measuring? 

20. In 316754 Ibs. of hemp, how many tons, cwt., etc.? 
Answers: 141, 8, 18; 17160; 3, 8; 66| ; 945; 27690; 59, 

9, 25, 4, 59; 49|; 475, 7, 9; 7jf ; 26.48 T \; 1497|; 1632960. 
114. To add compound numbers. 

1. What is the amount of the following sums of British 
money ? 

s. d. SOLUTION. 1. We first add the fractions, calling 

18 17 4^r them farthings, which makes 6 farthings; these we 

19 g 7JL reduce to pence by dividing them by 4. f If o p 

17 7 83- *-' Wr * te ? ancl a(1( * tlie * P ennv to tn e column of 

5 pence, which makes 20 pence; this number divided 

55 11 8-J- by 12 (the number of pence in a shilling)^! shilling 
and 8 pence. Write the 8 under the pence, and add 1 to the units 
of the shillings' place, which makes 21; write 1 and add the 2 to 
the ten's column=3 or 31 shillings, which, divided by 20 1 and 
11 shillings left. Write the latter under the shillings and add tho 
1 pound to the pound's column=r55. Ans. <55, 11s, 8J. 

Add the following: 

2. 17 18 llf + 14 17 2J+ 16 14 8 =? 

3. 17 19 OJ+ 45 11|+111 10 2J=:? 
t. 116 16 6 +320 14 5{-+ 38 18 8 =? 

5. 24 18 6 + 180 10 Of+ 66 19 11|=? 

6. 175 19 7|+ 90 8 8|+575 12 6=? 

7. 201 17 6|+1010 10 10]+970 19 11|=? 
Totals, 3297 17 9f 700, 10s, 8d. 

Hhtftk^- 



COMPOUND NUMBERS. 121 

115. To subtract compound numbers. 
1. From 19, 4s, 4d take 14, 7s, 6]d. 

s. d. EXPLANATION. We can not take J from nothing, 
19 4 3 so we add a penny to both terms; subtracting -J- from 
14 7 6^ the 1 penny, or 4 fourths, we have Jleft. Adding Id 
^T"T^ ^ to the 6d we have 7d, which we can not subtract 
* from the 3d above, and accordingly add Is to both 
numbers. 7 from Is 3d or lod, leaves 8d. Adding Is to the shil- 
lings, we have 8s, which can not be taken from 4s without adding 
1 to both numbers; 1 to 4s=24s; 8s from 24s=16s. Then 
adding 1 to the 14, we have 15, which, taken from 19=4, 
making the answer 4, 16s. 8Jd. 






Subtract the following: 

s. d. s. d. 

2. 17 10 8i 14 53:=:? 

3. 119 7 6 17 19 5J=? 

4. 500 20 18 8 =? 

5. 176 14 7] 129 15 7%=? 

Total, 630, 13s, 9Jd 
116. To multiply compound numbers.* 

6. Multiply 17, 4s, 9J by 8. 

OPERATION. 17 4 9| 



137 18 2 

After performing operations in addition, the learner 
will readily see how this is done. 

7. 17 18 8J X 7==? 10. 48 9 6JX2 and 3=? 

8. 120 16 61X12=? 11. 145 8 

9. 365 7|X 9=? 12. 705 13 

Total, 4860, 15s, OJd. Total, 5639, 19, 6. 

* Multiplication may likewise be performed by reducing the com- 
pound number to one denomination. (See Reduction) 
11 



122 NELSON'S COMMON-SCHOOL ARITHMETIC. 

117. When the multiplier exceeds 12 and is a composite 
number, or otherwise. 
13. 48, 9, 



s. d. EXPLANATION. Here we multiply by the two 
48 9 6J factors of which the 24 is composed. 
3 



145 8 7^=3 times the amount. 



1163 9 =8 times 3 or 24 times the amount. 

14. Multiply 705, 13s, 9|d by 38. 
705 13 9|X2 



4234 2 10|=6 times the amount. 
6 



25404 17 3 =6 times 6 or 36 times the amount. 
1411 7 7^=2 times the amount. 

26816 4 lQ$=Sum 0/2+36 times or 3%Xthe amount. 

15. 19 6s 7 dX 84=? 18. 27 8s 8 dX 87=? 

16. 91 18s 5idX 89=? 19. 77 17s 7|dX 95=? 

17. 4s 7JdXl29=? . 20. 89 17s 6 dx!50=? 

21. 176 Xl7=? 24. 5 7 6|X26=? 

22. 349 X19=? 25. 638 X29=? 

23. 4 5 7^X23=? 26. 8 4 7|X30=? 

Answers: 23266, 15, 4|; 9834, 12, 5; 566, 1, 3; 
183, 7, 11. 

118. To divide compound numbers. 

British money being almost the only thing in business to which 
compound numbers is applied exercises in it have received most at- 
tention; and especially as direct importation gives the clerk more 
to do with it than heretofore. See index for shorter methods of 
computing this kind of money 



COMPOUND NUMBERS. 123 

27. Divide 157, 13, 6J, by 5. 

EXPLANATION. 5 is contained in 157 31 

s. d. times and 2 over. These 2 reduced to shil- 

5)157 13 6J- lings, and added to the 13s. of the dividend, 

31 10 8^- make 53s., in which 5 is contained 10 times 

and 3s. left. In 3s. there are 36 pence, which, 

added to the 6d. of the dividend, make 42d., in which 5 is contained 
8 times and 2d. over. In 2d. and |d. there are 10 farthings, in 
which 5 is contained 2 times, making j or ^d. 

28. Divide 157, 13s, 6Jd equally between 25 persons, 

25) 157, 13s, 6Jd(6, 6s, IJd, or 6, 6s, l|d, nearly. 
150 

7 =remainder. 
20 



W3=shilUngs in 7, with 13s of the dividend added. 
JL50 

3=remainder in shillings. 
12 



42=pence in 3 shillings and 6 pence from the div d. 



Yl=remainder in pence. 
4 



70=/ar things in 17 pence and \. 
50 



20=remainder) or |-5=-f farthings. 

s. d. s. d. 

29. 487 13 -r- 9=? 32. 167 18 6|-5- 25=? 

30. 356 7 10 -r-36=? ' 33. 768 14 3 J -5-125=? 
*31. 419 15 6-r-14=? 34. 17 11 3|-r-875=? 

Answers : 12, 17, 8|, and 94, 1, 4|. 

* When the remainder from farthings is J or over, add a farthing, 
otherwise omit it* 



124 NELSON'S COMMON-SCHOOL ARITHMETIC. 



GENERAL EXERCISES IN COMPOUND NUMBERS. 

28. In 4 yards 2 feet seven inches, linear measure, how 
many inches? 

29. In 100 inches how many yards? 

30. Reduce 3520 yards to miles. 

31. In 100,000 inches how many miles? 

32. How many times will a carriage-wheel turn in a 
distance of 17 miles, the wheel measuring 2 yards 2 feet 
in circumference? 

33. From a plank 17 yards long was cut 10 yards 2 
feet 3 inches; how much of it was left? 

34. Reduce 3 acres 140 rods to square yards. 

35. Divide 200 acres 100 rods into 10 equal parts. 

36. From 406 acres 17 rods take 68 acres 148 rods 15 
yards. 

37. Divide 64 acres 134 rods 8 yards into 5 equal parts. 

38. Find the price of 9| ounces of gold at 3 17s 8d 
per Ib. Troy.* 

39. Find how often 2 4s 6d is contained in 41 10s 
7>d. 

40. Reduce 1 Ib. 3 oz. 5 pwt. to pennyweights. 
Answers: 175; 11220; 1, 4, 24, 5, 2, 4; 2; 2, 2, 4; 6, 

9; 337, 28, 15} ; 2, 19, 10^; 12, 3, 34, 25; 18|||; 18755; 
20, 10 ; 305. 

The Teacher should require his scholars to give the denomination 
of each item in the answers. 

*Find the price of 1 ounce j then the price of 9J. 



SHORT METHODS OF MULTIPLYING. 125 



XIII. SHORT METHODS OF MULTIPLYING. 

119. BESIDES the contractions by aliquots, under Art. 
66, the expert accountant and arithmetician can find ab- 
breviated methods adapted to almost every calculation. A 
few will be given in this place, to admit of the learner using 
them, when opportunity OCCIMTS, in the subsequent exercises. 

120. To multiply by 11, write the first figure of the 
multiplicand as the first of the product, and add each 
figure on the left to the one on the right, as below. 

1. 38397X11=427867. 

Prove the following by multiplying in the ordinary way : 

2. 379X11=? 7. $219.168X11=? 

3. 1487XH=? 8. $716.573X11=? 

4. $37.486X11=? 9. $316.144x11=? 

5. $9314.20 XH=? 0. $137.211X11=? 

6. $167.473X11=? 11. $710.22 XH=? 

121. To multiply by the teens when the tables are not 
known, and by such numbers of two figures as end with 1, 
as 21, 31, etc., multiplication by the figure 1 ought to be 
omitted. 

12. Multiply 3174 by 17. 

3174 EXPLANATION. The product of 7 is written one place 

22218 to the right to allow the first line to stand in the tens' 
place, by which it is multiplied by 10. 



13. $3163 X15= ? 18. $435.16fXlG=? 

14. $216.37 X19=? 19. $213.14 Xl8=? 

15. $1139.24 X13=? 20. $1137.37^X19=? 

16. $413.22 X18=? 21. $713.1Hxlti=? 

17. *8131.18|X14=? 22 - $4302.87 X 17=? 



*Call the J of a cent 25 hundredths, making 1311825 for the 
multiplicand, and point oft' four figures. (See note, page 06.) 



126 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



23. $316.27X51=? 

31627 EXPLANATION. The 5 of the multiplier being in the 

158135 tens' place, the first figure of the product -is written 
if 19 077 under the tens of the multiplicand. 



24. $137.50 X31=? 

25. $298.67 X&1=? 

26. $783.37JX61=? 

27. $313.17 X81=? 

28. $1987.871X91=? 




29. $2136.22 X 71=? 

30. $1394.311 X 41=? 

31. $653.18|X 21=? 

32. $291.16* Xl21=? 

33. $312.18|X 21=? 
122. When the multiplier wants from 1 to 12 of being 

100, 200, 3000, etc., the work may be contracted by mul- 
tiplying by one of these, and subtracting as many times 
the multiplicand as the multiplier is short of it. 
34. To multiply 424 by 97. 

OPERATION. 424x100=42400 
424X 3= 1272 



35. 765X192=? 

30. 1789X398=? 
37. 67.84X188=? 
9876X191=-? 
671 X B9=? 
59X689=? 



38. 
39. 
40. 



41128 

$89 
$167 

$37.98 
$478.96 



X784=? 
X 29=? 
X489=? 
X499=? 






41. 
42. 
43. 
44. 

45. $674.82^X992=?' 

46. $7164.37-^X 87=? 

123. When the multiplier is 29, 39, 49, etc., we multi- 
ply by the next higher number and subtract the multipli- 
cand, 

47. To multiply 176 by 59. 

OPERATION. 176X60=10560 
J 176 . 

10384 

124. To multiply ly 601, 1003, 90001, etc. 

The case differs from Art. 8 only in the intervening 



SHORT METHODS OF MULTIPLYING 127 

figures; so the product is written one place further to the 
right or left for every cipher. 

^48. Multiply 317 by 601. OPERATION. 317 

1902 

Ans. 190517 

49. Multiply 15704 by 10007 OPERATION. 15704 

109928 

Ans. 157149928 

125. When one part of the multiplier contains the other 
without a remainder, as 248. Here 24 contains 3 times 
the 8 or first figure, so by multiplying the product of 8 
times the multiplicand by 3, one line is saved. 

50. Multiply 76439 by 248. 
OPERATION. 76439 

611512=8 times 76439 
1834536_=3 " 611512 

18956872 Ans. 

REMARK. This operation might be shortened by multiplying the 
product by 8 mentally, and adding that line for the whole product. 

51. Multiply 25938 by 936. 

OPERATION. 25938x936 or _ 25938 

233442 ~~ 233442.. 
933768 



24277968 



24277968 

52. 11457X324=? 57. 7832x64256=? 

53. 672x189=? 58. 7498x16144=? 

54. 783x357=? 59. 9739X 3972=? 

55. 924X218=? 60. 6487X 8109=? 

56. 596X426=? 61. 74675X 7206=? 

126. To multiply l>y 375, 625, 750 or 875, we first mul- 
tiply by 125 (Art. 66), and that product by 3, 5, 6 or 7, 
these numbers, 375, etc., being multiples of that number. 



128 NELSON'S COMMON-SCHOOL ARITHMETIC. 

1649000X5 8245000 

62. 1649X625= g^-= g 1030625 

63. 3156X375=? $1703.20x750=? $1456x875=? 

127. To multiply by a composite number composed of 
factors under 13, the latter may be used as multipliers 
instead of the former. 

64. 314X72. 

OPERATION. 314x12=3768x6=22608. 

65. 932X64=? 68. $913.27 X^4=? 

66. 738X48=? 69. $293.75 X72=? 

67. 426X96=? 70. $6318.371x63=? 

128. To multiply by any number of 9s, we multiply by 
the next highest number and subtract the multiplicand. 

71. 3145X999- 

3145X1000=3145000 
Prom which take 3145 

3141855 

129. To square, mentally, numbers under 39 that end 
with 9. 

72. What is the square of 29? 

29 EXPLANATION. Writing 1 for the first figure of the pro- 
29 duct, we add 1 to the tens' place of the multiplier, and rnul- 
7777 tiply the sum on the multiplicand less 1 : 3X28=84, with 
the 1 annexed=841. 

73. Find the square of the following numbers mentally: 
99, 59, 119, 79, 19, 69, 129, 89. 

130. To square any number of 9s instantaneously, and 
without multiplying. 

Commencing at the left, we write as many 9s, less one, 
as the number to be squared, an 8, as many Os as 9s and 
a 1. 

74. The square of 9999999 is 99999980000001. 



SHORT METHODS OF DIVIDING. 129 

The square of any number of 3s will be one-ninth of 
the square of the 9s. 

131. To square numbers under 135 ending with 5. 

The first two figures on the right of the product will 
always be 25 ; and to find the others, we add 1 to the 
tens' place and multiply it on the tens' and hundreds' 
places above. 

75. To square 115. OPERATION. 11 

12 

13225 

The reason of this will be apparent by multiplying in 
the usual way. 

132. To square a number containing a half, as 12J, we 
multiply the whole number by the next higher number 
and add a fourth. 8| squared=8x9-|-{=72J. 

76. Find the square of the following numbers: 99999, 
33333, 75, 45, 65, 62 16J, 19J. 



XIV. SHORT METHODS OF DIVIDING* 

133. Division may often be contracted by cancellation* 
tvhen the terms are written in fractional form. 

1. Divide 1463 by 28. 

209 EXPLANATION. The terms 1463 and 28 were first 

divided by 7, leaving 209 fourths, and 209 divided by 

* 4 S ives 52 i- 



To THE TEACHER. The author does not offer all these contrac- 
tions as rules of general utility; still, he is of opinion that a fa- 
miliar knowledge of them will be advantageous to the student of 
arithmetic in disciplining his mind and showing him the relation of 
numbers. Where the instructor thinks otherwise, he can omit them. 

* To cancel signifies to blot out or make void. 



130 NELSON'S COMMON-SCHOOL ARITHMETIC. 

Prove the answers obtained, by dividing in the usual 
way. 

2. 3465-v-35=? 2763-4- 81? $65.45--243= ? 

3. 1962-r-22=? 6876-r-152=? $54.36-r-144=z? 



134. To divide by aliquots of 100, 1000, etc. 
This process is the reverse of that under Art. 66. 

135. To divide by a composite number, as 96, which is 
composed of the factors 12 and 8, or 648, which is com- 
posed of 9X8X9. This operation is performed by using 
the factors instead of the whole number. 

4. Divide $78.54 by 32. OPERATION. 4)7854 

8)19632 

2453 

How the true remainder is found: 

The first remainder is 2 cents, because it is left from the 
cents that were divided. The second remainder is four 
times as great as if it were from the first line, because 
every figure of the second line is four times as great as if 
it stood in the first line. Four times 312 and the 2 of 
the first remainder equal 14, the true remainder. 

Ans. 245f . 

5. Divide 6371 by 336. OPERATION. 6)6371 

7)10615 

REMARK. The true remainder of this example is - ~~ 
found by multiplying the last remainder by 7, to '_l__ 
make it of the same value as if it were from the line 18 7 

above, and that by 6, to make it of the same value as if it were 
from the upper line: 7X?X G:=r: 2H to which add 6X4+5 or 29. 
The true remainder is 323. Ans. 1 8|||. 

6. Divide 1463 by 28 10. 4571-f-441==? 

7. " 7614 " 72 11. 1987-^379=? 

8. " 1943 " 49 12. 9843-^-720==? 

9. " 8765 " 343 13. 1456-f-729=? 



SHORT METHODS OF DIVIDING. 131 



14. Divide 7654 by 25. 



76.54=10<M of 7654. 
4 



306.16=4 times as much, or ^ of 7654. 
or 306^=306^. 

15. $3675-^-125 = ? S213.67-s-16=? $11 74-4-121=? 

16. $2153-- 33$=? $319.25--- 8$=? 316-^25 =? 

136. When the divisor is 15, 35, 45, 55 or 65, it will 
abbreviate the work to multiply the dividend by 2 and di- 
vide by 30, 70, etc. 

17. 345-^-35=? 345x2=690-^-7=98f 

18. 2756-4-15=? $1324.25-4-35? $365.75-4-45=? 

137. To divide by 75, 175, 225 or 275, the dividend 
may be multiplied by 4 and the product divided by 300, 
700, 900 or 1100. 

19. 2136-4-75=? 

2136 EXPLANATION. 4 three-hundred ths being equal 
4 to 1 seventy-fifth, it abbreviates the work to divide 
3 00^)85 44 b ^ ^^ an( * mult *P 1 y by 4. 



20. 3678-4-175=? 6317-r-175=? $19.32-f-275=? 

138. Long division may be abbreviated by performing a 
part of the process mentally, and writing only the result. 

21. 76354-^-34=? 

34)76354(2245 EXPLANATION. The products are omitted; only 
83 the remainder and the figures brought down are 

155 written. 

194 
24 

22. 3167^-184=? 1679-i-21=? 67831-4-498=? 

139. To divide by any number of 9s, the dividend may 
be pointed off, from the left, in periods corresponding with 



132 NELSON'S COMMON-SCHOOL ARITHMETIC. 

the number of figures in the divisor. If the divisor have 
two 9s, it may be pointed off in periods of two; if three 
9s, in periods of three, and the operation performed as in 
the formula. 

23. 316328-;-99^? 7312961--999=? 

31,63,28 731,296,1 REMARK. It will be observed that 

31,63 731,2 each line in the formula is diminished 

31 7 by one period, corresponding with the 

01 OR oo ^7OOA OQA number of 9s, and the sum of the whole 

oiyo.-u.iu lo^u.^oU , . ., p. , , . 

1 -. taken. As many figures as Us should 

I ~ be pointed off for a remainder, plus 

3195.23 7320.281 the carrying figure of the first line. 

24. Divide 167389 by 9999 and 7654321 by 99.* 



XV. MARKING GOODS GAIN AND LOSS. 

140. ON receiving goods for sale, the merchant or 
some of his clerks 'examine them by the bill or invoice, 
which is usually received in advance of them by mail, 
after which they proceed to mark them. 

141. Marking Goods. This is done by selecting sam- 
ples of each kind or quality of goods, and putting on 
them a private mark, indicating the cost price, the selling 
price, or both. 

Every house has its own private mark, which usually 
consists of a word or phrase to represent the ten digits, 
as the word importance, which has ten letters. Commenc- 
ing at 1, the letters are arranged as follows: 

importance 
12 34567890 
In addition to these, another letter not contained in the 
above is selected for a repeater, so that when a figure oc- 

* It would be no abbreviation to divide by 9 in this way. 



MARKING GOODS GAIN AND LOSS. 133 

curs twice it may be inserted to prevent detection. The 
letter g will do in this case. 

To mark *6.55, the letters trg would be used, with per- 
haps a line or period to separate the dollars from the cents. 

To fix on a selling price, various circumstances have to 
be taken into consideration-^-the cost of transportation, 
the probable length of time required to sell, cost of rent, 
wages of clerks, depreciation of stock, etc. 

A certain rate is put on for the first item, say 5 to 15 
per cent.; then an amount estimated to cover the balance 
and leave a profit. When adding the entire rate of ad- 
vance to the cost price, the clerk is not required to be 
exact, as simplicity of calculation is a greater object than 
uniformity of profit. Thus, in marking goods to sell by 
the dozen, a multiple of 12 would be preferred, whether a 
little above or below the fixed rate. For the same reason, 
aliquots, or numbers formed of aliquots, of 100 would be 
selected. 

When working the following exercises, the learner will 
remember that fractions, unless occurring with aliquots of 
100, are usually omitted. When under ^ they are rejected; 
otherwise, a cent is added to the cents/* 

1. Add 25% to $3.50, $5.75, $1.82, .75, $2. 
Answers: $4.37 2 , $7.18f, $2.28, .94, $2.50. 

2. Add 16|- % to $6.20, $3.12 2 , 27c, $3.87. 

3. Add 37% to 20c, 12 2 c, $1.15, 6c, 15c. 

4. Add 15% to $1.20, 75c, $1.22, $3.57, 16 2 c, 27c. 

5. Add 33J% to 5c, 15c, 21c, $6.75, 18c, $27.50. 
Answers: $15.72, $8.24, $46.46, $2.32. 

*To indicate the degree of exactness required, a few answers will 
be given to the examples which follow. 

One object in giving these examples being to exercise the judg- 
ment of the pupil, he will not be required to obtain precisely the 
same figures as given in the other answers. 



134 NELSON'S COMMON-SCHOOL ARITHMETIC. 

6. $1.75+14% = ? 14. $6.75+26%=? 

7. .87+12% = ? 15. .07+40% = ? 

8. $1.67+53% = ? 16. $3.60+12% = ? 

9. .04+60% = ,? 17. $4.80+30% = ? 

10. $2.75+35% = ? 18. $1.20+35% = ? 

11. .16+15% = ? 19. $7.20+40% = ? 

12. .05+25%=? 20. $6.00+14% = ? 

13. $3.14+18%=? 21. $5.10+25% = ? 
Answers: $13.25, $43.79. 

22. To .30 add 20 % profit. 

23. To $1.20 add 5 % charges and 20% profit. 

24. To $1.75 add 2J% freight and 10% profit. 

25. To .08 add 16| % profit. 

26. To .36 add 33J % profit. 
Answer: $4.41. 

27. $0.10+10%=? 30. $1.00+16|%=? 

28. .05+50%=? 31. $3.25+334% = ? 

29. .25+20%=? 32. $1.40+12^%=? 
Total, $7.57. 

The following may be worked by aliquots of 100: 

33. Add 5 % to $12.50 44. To $3.50 add 4 % 

34. " 6|-% " $7.80 45. " $9.50 " 

35. " 84% $4.85 46. $0.10 " 

36. " 9 % " $3.50 47. " $0.18f " 

37. " 124% " $0.87 48. " $0.06^ " 40 % 

38. " 20 % " $0.45 49. " $0.87^- " 50 % 

39. " 18f% $0.15 50. " $0.084 75 % 

40. " 25 % $0.124 51. " $0.11J" 20 % 

41. " 7 % " $6.20 52. " $0.80 " 

42. 34% " $1.00 53. " $0.16 " 

43. " 12J>% " $1'.35 54. $0.47 
Answers: 41.53, $18.56. 



GOODS GAIN AND LOSS. 



135 



ORAL EXERCISES .* 



1. $0.10+ 25 % = ? 


18. $0.50 + 20 % = ? 


2. .25+ 10 % = ? 


19. .25 + 50 % = ? 


3. .20+ 6 % = ? 


20. .11 + 15 % = ? 


4. .33+ 33J% = ? 


21. .12^+ 20 % = ? 


5. .16+ 25 % = ? 


22. .45 + 9 % = ? 


6. .45+ 5 % = ? 


23. .66 + 33|% = ? 


7. .87+ 30 % = ? 


24. .14 + 5 % = ? 


8. .16+ m%=? 


25. .35 + 20 % = ? 


9. .10+ 22 %=*\ 


26. .16 + 50 % = ? 


10. .05+ 30 % = ? 


27. .67 + 12 % = ? 


11. $2.20+ 50 % = ? 


28. $1.20 + 16|% = ? 


12. $550 + 100 %=? 


29. .27 + 50 % = ? 


13. .75+ 60 % = ? 


30. $6.50 + 16 % = ? 


14. $1.25+ 30 % = ? 


31. $2.15 + 8 % = ? 


15. $1.12+ 25 % = ? 


32. $1.87 +200 % = ? 


16. .75+ 20 *T=? 


33. .31 + 18 % = ? 


17. 1.50+ 8 % = ? 


34. .19 + 12% = ? 


35. 30% of $20.00=? 


43. $0.37^+30 % = ? 


36. 50% of $1000.00==? 


44. .62J+ 5 % = ? 


37. 25% of $700.00 = ? 


45. $1.25 +12J% = ? 


38. 16% of $500.00=? 


46. .16 + 5 % = ? 


39. 10% of $350.25 = ? 


47. .54 +40 % = ? 


40. 15% of $200.00=? 


48. $30.00 +60 % = ? 


41. 80% of $500.00=? 


49. $90.00 + 9 % = ? 


42. 20% of $20.00=? 


50. $70.00 +30 % = ? 



* The Teacher may require the flections ill these exercises. 



136 NELSON'S COMMON-SCHOOL ARITHMETIC. 



GAIN AND LOSS. 

142. Merchants distinguish between real gain or loss 
and gain or loss per cent., calling the former the actual 
gain or loss, and the latter the gain or loss per cent. 

143. To find the actual gain, it is simply necessary to 
subtract the cost price from the selling price. 

1. Bought a house and lot for $4367 and sold them for 
$5000; how much did I gain? 



=(7os or buying price. 
$633=:-4c2waZ gain. 

Cost price. Selling price. Cost price. Selling price. 

2. $2.75 $4.87 7. $316.17 $215.25 

3. $97.35 $120.10 8. $112.14 $120.48 

4. $6.87 $6.98 9. $317.18| $21.9.m 

5. $5.40 $9.80 10. $67.21 *8&2J8 

6. $3.20 $6.40 11. $54.12J 821.18J 
Total gain, $32.58. Net loss, or loss with gains de- 

ducted, $204.51. 

144. To find the gain per cent., is simply to find the 
gain on every hundred dollars or cents. 

Required the gain per cent, on goods which sold at 
$1.35 and cost $1.20. 

135120=15 cents, the actual gain on 120. 15-=- 
120= T yg=gain on 1 cent. y'&X 100=^^=12 J per 
cent., or gain, on 100 cen-ts. 

OPERATION. 135 EXPLANATION. The actual gain 

120 is first found; then the gain per 

cent., by dividing the actual gain 
( when multiplied by 100) by ti& 
first cost. 



MARKING GOODS GAIN AND LOSS. 137 

12. Goods which cost $2.00 were sold at $3.00; required 
the gain per cent. 

13. The cost price was $1.25; the selling price, $1.50; 
what was the gain per cent.? 

14. Goods bought at 75 cents sold at $1.00; what was 
the gain per cent. ? 

15. 10 cents was the cost; 12 \ the selling price; re- 
quired the gain per cent. 

Total rates of gain of the four, 



First cost. Selling price. First cost. Soiling price. 

16. $12.50 $10.00 20. $3167.00 $3000.00 

17. .18 .20 21. $1000.00 $1500.00 

18. .05 .06 22. $27.80 $20.00 

19. $127.52 $111.58 23. $12.17 $11.50 
Net loss, on 16 to 19, \-^%. Answers to second group, 



24. Bought a bbl. of apples for $1.75 and sold it for 
$2.25; what did I gain per cent.? 

25. Sold 25 bbls. of potatoes for $39.00; how much did 
I gain per cent., if they cost me $1.25 per barrel? 

26. Bought 150 bbls. of flour @ $5.25, paid for dray- 
age $7.50 and porterage $1.00; at what per barrel should 
I sell it to gain 15 per cent.? 

27. Bought 15 horses at $125 each, and sold the lot for 
$3500; what was my gain per cent., after paying $25 for 
their feed? 

28. Sold a safe which cost me $80 for $75; what was 
my loss per cent.? 

29. Bought a bill of goods for $350, paid freight $15.20, 
insurance $5, drayage $3, and sold them for $425; what 
was my actual gain, and what my gain p^er cent.? 

30. Sold A's note for $750 at a discount of 15%; what 
did I pay for it? 

12 



138 NELSON'S COMMON-SCHOOL ARITHMETIC. 

31. Sold B's note of $320 for $300; what was the rate 
of discount? 

Answers: $637.50, $51.80, 14% nearly, 84J% nearly, 
6 i%> 6 f%> 28 T%> 24 f%> $ 6 - 10 - 

145. When the selling price and the rate per cent, are 
known, to find the first cost. 

32. What was the first cost of goods marked $2.65, the 
rate per cent, of profit being 25? 

1.25)2.65(212 EXPLANATION. Every dollar invested in 
2 50 or $2.12 the goods has increased 25 cents, and is 
- worth 125 cents. Hence, there are as many 
irQfl invested dollars in $2.65 as 1.25 is con- 
tained times in it. The remainder, 15 dol- 
lars, we reduced to cents, which, divided by 1.25, gives 12 cents. 

Another way: Since 25% is J of 100, the $2.65 must 
be I more than the first cost. Let the first cost be f, 
then J+|=|. Therefore, $2.65=f of the first cost. 

2|5. = 53=one-fourth of the first cost. 
53X4=2.12, first cost. 

What was the first cost of the goods marked as follows? 
The learner can prove his calculations by reversing the 
process. 

33. $2.25 @ 10 % gain 38. $2.87 @ 10 % loss 

34. $3.70 " 5 % " 39. $1.54 " 6 % " 

35. $115.87 " 12%% " 40. $3.75 " 25 % " 

36. $14.54 " ^1% " 41. .87J " 12%% 

37. .87 " 16f% " 42. .12% " 5o"'% " 

43. $9.50 @ 50 % gain 47. $90.00 @ 20 % gain 

44. $7.87 " 25 % " 48. $75.30 " 15 % 

45. $6.50 " 16f% " 49. $82.50 ' 

46. $8.75 18f % " 50. $60.00 20 



COMMISSION AND BROKERAGE. 139 



XVI. COMMISSION AKD BROKERAGE* 

146. COMMISSION, or brokerage, is the percentage 
charged by a commission merchant, factor, agent or broker 
for transacting business for another. 

147. Commission is usually reckoned on the whole 
amount of sale, purchase or collection. 

1. At 2J per cent., what is the commission on $17640? 

Ans. $441. 

2. A merchant sells goods for another to the amount 
of $4371.81; what is his commission at 5 per cent.? 

3. A broker receives | per cent, for selling $2500 worth 
of merchandise for a commission merchant; what is the 
amount of his brokerage? 

4. A of New Orleans buys sugar for B of Cincinnati to 
the amount of $7100; what is the amount of his commis- 
sion at 1J per cent.? 

5. A commission merchant sells goods for his principal 
to the amount of $3000, and charges 2^ per cent, commis- 
sion; what does he make by the operation, after paying a 
broker | per cent, for his services in effecting sales? 

6. After receiving 5 per cent, commission on sales 
amounting to $520.75, how much should I return to my 
principal? 

7. Gave a lawyer a note of $50 to collect, at 8 per cent. ; 
how much should I receive? 

Omit fractions of a cent in the answer: 

Answers: $106.50, $6.25, $218.59, $67.50, $494.71, $46, 
$510.71. 

; 

*See Commission Merchant, page 94; Brokers, page 95. 



140 NELSON'S COMMON-SCHOOL ARITHMETIC. 

What is the commission on the following amounts? 

8. $364.15 @ 3$i = ? 12. 36.21 @ 

9. $78.54 6^% = ? 13. $174.09 

10. $710.06 83% = ? 14. $2167.90 

11. $876.75 " 9J%r=? 15. $78.21 " $% = 1 

Total,*$161.31. Total $146.13. 

16. A commission merchant charges 2J% com. and 2^r% 
guarantee on the sale of goods amounting to $3100; how 
much should he return to his principal? 

17. A merchant receives a consignment of goods valued 
at $3000, and sells them for $4500 ; how much should he 
remit to the consignor, after reserving 4J% for com. and 
guar. ? 

Answers: $2945, $4297.50, $5297.30. 

18. A merchant sells a note of $100 to a money broker, 
at a discount of 6 per cent.; how much money does he 
receive? 

19. A New York merchant buys a bill of exchange 
worth $400, on a Cincinnati banking house, at J per cent, 
discount; how much does he pay for it? 

20. Bought a bill of exchange on New York for $7691, 
at l^r premium; what did I pay for it? 

21. Purchased 20 shares of railroad stock, worth $20 
per share, @ 12^% discount, and sold it at par; what was 
the amount paid? and how much did I gain? 

Answers: $94, $399, $7806.37, $350, $50. 

22. "What is the premium on the following? $31.46@ 
5J%; $1760@6i%; $4617@9i%. Total, $550.35. 

148. To find tlie commission on investments. 

The merchant often has moneys in his hands or re* 
mitted to him for investment in goods or stocks, upon 
which he is allowed commission on the amount invested 
only. 



COMMISSION AND BROKERAGE. HI 

23. At 2-|-% commission, what amount of money shall I 
retain of $2000 in my hands for investment? 

This $2000 contains 100% of the amount to be invested, 
plus 2.5%, my commission making 102,5% of it. 

Reducing both to tenths, we have 2000.0 to be divided 
by 102.5. 
102.5)2000.0(19.5121 or $19.5122=1%, 

1025 Which, multiplied by 2J gives the com- 

9750 mission, or by 100, gives 100 per cent., or 

0995 the amount to be invested. Com $48.78. 

5250000 REMARK. The 525 dollars remainder were 
5125 reduced to tenths of mills. 

125<T 
1025 

2250 

2050 



2000 
1025 

975 

RECAPITULATION. To find 1 per cent., we divide 2000 by 102.5, 
which, multiplied by 100, gives 100 rer cent., or the amount. The 
difference is my commission. 

24. What will be my com. on $1300, to be invested at 2%? 

25. At 5% commission, how much sugar can I buy for 
$3475.25, when it sells at 20 cents a pound? 

26. Out of $987.50 how many pounds of tobacco should 
I purchase for my principal at 35 cents a pound, commis- 
sion 2|%? 

27. At 2|-% commission, how much shall I reserve of 
$2130.67, after investing in cotton at 25 cents a pound? 

28. To pay 12-J- cents a pound for rice, and -fa% for 
insurance, and myself 3% commission, how many pounds 
can I buy for $345.15? 

The learner will prove his work by reversing the process. 



142 NELSON'S COMMON-SCHOOL ARITHMETIC. 

29. Invest $1367.37, \ in gloves at 30 a dozen, \ in 
sheeting at 35 cents a yard and the balance in muslin at 
20 cents a yard; commission 2J%. 

INTEREST. 

149. Interest is a percentage allowed for the use of 
capital, which may consist of money, merchandise or debts 
due. It is regulated by the year or month. 

150. The sum upon which interest is reckoned is called 
the principal; the percentage allowed per month or year, 
the rate; and the sum of the principal and interest, the 
amount. 

151. Interest is divided into simple and compound 
Simple interest is percentage on the principal alone; com- 
pound interest is interest reckoned on both principal and 
interest. 

The legal rates of interest in the several States is vari- 
able. 

152. Money loaned on interest is usually secured by 
promissory notes, expressed as follows: 




153. This note would be due on the 1st day of Decem- 
ber but for a law which allows three days longer for its 
payment, called days of grace. It will therefore be legally 
due on December 4th, and interest will be reckoned to 
that date. 

* Other forms of notes will be found under Banking. 



SIMPLE INTEREST, 143 



XVII. SIMPLE INTEREST. 

154. Interest is usually calculated on the basis of 360 
days to the year.* When notes are drawn by the month, 
calendar months are understood. Thus, a note drawn on 
the 1st of September, as the preceding, falls due on the 
1st of December, plus the days of grace. 

155. The simplest method of computing interest is to 
do it at the rate of 6 per cent, per annum, and add or 
subtract when it is higher or lower than that. 

156. The interest for 60 days at 6% per annum is 
equal to as many cents as there are dollars, or, in other 
words, is 1 per cent, of the principal. 

The reason of this is obvious. 6 per cent, per annum 
is ^ per cent, per month, or 1 per cent, for two months or 
60 days. 

ORAL EXERCISES. 



1. The interest for 60 days 
2. " " " 60 " 
3. " " " 60 " 
4. " " " 60 " 
5. " " " 60 " 
6. " " " 60 " 
7. " " " 60 " 
8. u " " 60 " 


nt (\C/ 

, cLL \J yfl , 

u 6^ 
" 6&/ 
" 6 C / 
" / 

" 6^ 

" *; 


on $20 =? 
$50 =? 

" $100 =? 
$75 =? 
" $125 =? 
" $175 =? 
" $200 =? 
" $316.50=? 


9. " " 


" 60 <( 


" ^9^ 


" $715 =? 


10. " " 


" 60 " 


" 6^! 


" $50.50=? 


11. " " 


" 60 " 


" *%> 


" $215.15=? 


12 u u 


" 60 " 


" 6% ' 


" $1000 =? 



*In New York interest is usually reckoned for the full year. 



144 NELSON'S COMMON-SCHOOL ARITHMETIC. 

13. The interest for 60 days, at 6%, on $976.14=? 

14. " 60 " 6%, $715.15=? 

15. " " 60 " " 6%, " $5000 =? 

16. " " " 60 " " 6%, " $5 ==? 

17. " " " 60 " " 6%, " $23.13=? 

18. " " 60 " " 6%, $67.15=? 

19. On $7.50=? 28. On $269.14=? 

20. " $8,75=? 29. " $198.97=? 

21. " $118.67=? 30. $267.18=? 

22. $368.56=? 31. $1365.50=? 

23. " $210.33=? 32. c< $316.18=? 

24. " $67.67=? 33. " $215.16=? 

25. $39.37=? 34. " $716.16=? 

26. " $21.37=? 35. " $317.60=? 

27. " $116.16=? 36. " $167.37=? 

157. Having the interest for 60 days, the interest for 
any shorter time may be found by 

ALIQUOTS OF 60. 

30 days=l 12 days= i 5 day 3=^ 2 days=^ 
20 ' = 10 = J 4 =^ 1 day =^ 
15 =1 6 = T V 3 = 5 V 

When the number is not an aliquot of 60, 

For 7 take 6 and 1 For 29 take 1 off 30 

" 8 " 6 " 2 35 " 30 and 5 

" 14 12 <c 2 " 38 " 30, 6 and 2 

" 19 15 " 4 43 30, 12 1 

" 26 " 20 " 6 " 45 15 oif 60 

" 27 " 15 " 12 85 add 20 and 5^ 

*The Teacher should continue these exercises till his scholars are 
familiar with all the aliquots, and the method of resolving other . 
numbers into aliquots. 



SIMPLE INTEREST. 145 

1. Fin.d the interest cm $500, for 30 days, at 6% per 
annum. 

3Q=)5.QQ=lHterestfor 60 days. 
2M=Interest for 30 days. 

2 to 8. Find the interest on $200 for 12 days, 15 days, 
20 days, 10 days, 6 days, 3 days, 1 day, at 6% per annum. 

AHS. $2.23.3. 

9 to 13. Find the interest on $160 for 1, 2, 3, 4 and 5 
days, at 6% per annum. Am. $0.39.9+. 

14 to 18. Find the interest on $240 for 6, 10, 12, 20 
and 30 days, at 6% per annum. Ans. $3.12. 

19 to 22. Find the interest on $1000 for 1, 10, 12 and 
6 days, at 6% per annum. Ans. $4.83.3+. 

23. Find the interest on $675, for 27 days, at 6% per 
annum. 

$6.75 =Iutcrest for 60 days. 

15=1= 1.687 
12=4- 1?5_ 

3.037 or $3.04. 

Principal. Time. Principal. Time. 

24. $250 for 20 ds=? 30. $650 for 35 ds=? 

25. $567 for 14 ds=? 31. $980 for 80 ds=? 

26. $968 for 25 ds=? 32. $216 for 93 ds^? 

27. $846 for 33 ds=? 33. $800 for 67 ds=? 

28. $610 for 18 ds=? 34. $915 for 44 ds=? 

29. $918 for 27 ds==? 35. $1200 for 93 ds ? 
Answers: $16.803+ and $54.448+. 

158. Merchants or bankers seldom reckon interest on 
cents. When under 50 they are rejected; otherwise, a 
dollar is added to the dollars. 

It should also be observed that business men express 
their results in dollars and cents, to which usage the 
learner ought to conform. For practice or review, the 
13 



146 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



51. 



teacher may require the exact answers instead of those 
given. 

40. $1799.14 for 93 ds=? 
$3$7.66 for 67 ds=? 
$199.44 for 41 ds=^? 
$450.22 for 29 ds^? 
Total, $35.75. 

48. $1997.00 for 13 ds=? 

49. $7.88 for 54 dsr^? 

50. $17.97 for 35 ds=? 
$10.00 for 120 ds^? 

Total, $4.71. 

58. $1999.20 for 23 ds=? 

59. $361.74 for 18 ds=? 

60. $78.93 for 23 ds=? 

61. $1467.20 for 34 ds=? 

62. $7100.18 for 77 ds=? 

63. $29.00 for 99 ds=? 

Total, $108.96. 

159. To compute 6% interest for any number of months. 

When used in drawing notes or drafts, the month is al- 
ways calendar; but when computing interest, 30 days is 
considered a month; hence, the note on page 142 would 
draw interest for one day more than the three months, 
exclusive of the days of grace, there being 94 days be- 
tween the date and the maturity.* 

This is a very important distinction, as will be seen by 
reference to Bank Discount. 

160. Since there are half as many 60 days as months, 
we multiply the interest for 60 days by half the number 
of months. 



36. $1000.00 for 27 ds=? 

37. $71.97 for 47 ds^=? 

38. $61.80 for 45 ds^? 

39. $190.27 for 16 ds=? 

Total, $6.04. 

44. $719.99 for 11 ds=? 

45. $55.18 for 9 ds=? 

46. $88.17 for 69 ds^? 

47. $466.00 for 78 ds^? 

Total, $8.47. 

52. $1000.00 for 97 ds=? 

53. $650.00 for 67 ds=? 

$10.70 for 13 ds=? 
$127.57 for 51 ds=? 
$368.17 for 118 ds^? 
$718.57 for 125 ds=? 
-Total. $46.76. 



54. 
55. 
56. 
57. 



*ln practice, when the note remains unpaid till maturity, interest 
would be charged for only the three mouths, as if it were 90 days. 



SIMPLE INTEREST. 147 

64. Find the interest on $620, for 4 months, at 6% per 
annum. 

Q.^= Interest for 60 days. 

2 Number of 60 days in 4 months. 

\2AQ=Interest for 4 months. 
Find the interest, at 6% per annum, on the following: 

65. $750.25 for 6 mos=:? 69. 910.70 for 5 mos=? 

66. $218.87^ for 8 inos=? 70. $876.33^ for 4 inos=? 

67. $1000.00^ for 7 mos=? 71. $937.79 for 3 mos=? 

68. $560.374 for 9 mos=? 72. $168.00 for 1 ino ==? 
Answers: $91.46 and $55.21. 

73. Required the interest on $350.25 for 7 mos. 15 ds* 

$3.50 =Interest for 60 days. 

&k=litumber of 60 days in 7 months. 

10.50 
1.75 
875=Interest for 15 days or J of 60. 

$13.125" or $13.12 2 . 
Another way: 

$3.50 ^Interest for 60 days. 



14.00 = Interest for 8 months. 
87 '5= Interest for 15 days off. 

13.125 or $13.12*. 
Compute the interest at 6% pr. an. on the following: 

74. * $36.57 for 3 mos 20 ds. 78. $1673 for 8 mos 8 ds. 

75. $2977 " 6 " 16 " 79. $936 " 4 " 19 " 

76. $9856 " 4 " 15 " 80. $281 " 3 " 27 " 

77. $2836 " 9 " 27 " 81. $106.27 " 8 " 16 lt 
Answers: $460.06 and $103.40. 

*When the principal is small and the time long, interest may be 
computed on cents. 



148 NELSON'S COMMON-SCHOOL ARITHMETIC. 

Find the amounts of the following, and be particular to 
add the cents of the principal : 

Mos. Days. Mos. Days. 

82. $250.15 for 6 2 86. $501.19 for 6 6 

83. $380.67 " 10 10 87. $219.12^ " 4 27 

84. $900.90 " 19 3 88. $369. 16f 8 12 

85. $216.67 " 8 20 89. $1220.00 10 10 
Answers: $1871.11 and $2408.91. 

161. To find the interest for years at 6% per annum. 

Business usage allows only 360 days to the year, which 
is 6 sixty days; hence, the interest for 60 days, multiplied 
by 6 times the number of years, gives the result. 

90. Find the interest on $120 for 1 year, 4 months and 
20 days, at 6% per annum. 

1.20 EXPLANATION. The interest for 60 days is 120 cents; 

8 for 1 year and 4 months it is 8 times 120 or 960 cents; 

q P~ and for 20 days it is J of 120, or 40 cents, making the 

' sum $10.00 the interest required. 

$10.00 Ans. 

91. Find the interest of $240 for 3 years, 4 months and 
10 days. Ans. $48.40 

92. What is the interest of $1467.45 for 2 years, 6 
months and 17 days? Ans. $224.21. 

Find the interest of the following: 

93. $321.00 for 2 years 3 months 15 days.* 

94. $1767.00 for 7 years 4 mouths 21 days. 

95. $897.25 for 3 years 6 months 27 days. 

96. $898.57 for 2 years 7 months 25 days.f 

97. $716.27 for 2 years 1 month 9 days. 
Answers: $90.57, $44.14, $783.66, $192.41, $143.09. 

*Find the interest for 2 years 4 months, and deduct the interest 
for 15 days. 

tCall this 2 years 8 months, and deduct the interest for 5 days. 



\ 



SIMPLE INTEREST. 149 

Find the interest on the following : 

98. $810.98 for 1 year 6 months 7 days. 

99. $50.00 for 9 years 7 months 18 days. 

100. 8.00 for 9 years 3 months 27 days. 
Answers: $90.58, $73.94, $10.90, $4.48. 

101. $3140.79 for 1 year 7 months 7 days=? 

102. $795.17 for 2 years 1 month 1 day =? 

103. $3.90 for 3 years 5 months 15 days=? 

104. $1057.57 for 1 year 11 months 11 days^? 
Total, $526.01. 

105. $2674.57 for 1 year 8 months 21 days=? 

106. $7143.45 for 2 years 1 month 18 days=? 

107. $1742.67 for 1 year 9 months 13 days=? 

108. $2100.00 for 2 years 1 month 1 day ==? 

109. $4109.85 for 1 year 6 months 17 days=? 
Total, $2022.35. 

110. $7856.00 for 1 year 1 month 29 days? 

111. $677.19 for 3 years 3 months 3 days=? 

112. $287.17 for 1 year 7 months 16 days=? 

113. $97.19 for 5 years 10 months 14 days=? 

114. $10.10 for 1 year 3 months 19 dys=? 
Total, $743.95. 

115. $57.87 for 2 years 6 months 14 days=? 

116. $120.14 for 7 years 7 months 7 days=? 

117. $340.00 for 9 years 1 month 24 days=? 

118. $1657.00 for 1 year 3 months 24 days=? 

119. $769.75 for 2 years 3 months 18 days=? 
Total, $487.40. 

. : . 

The Teacher, when reviewing these exercises, may require his 
class to compute interest on cents, as is usually done in courts of 
justice. 



150 NELSON'S COMMON-SCHOOL ARITHMETIC. 

162. Having the interest at 6% per annum to find it at 
any other rate, all-quo ts of 6 may be used. 

At 1% it will be of that at 6%. 

At 5% it will be -J- less than at 6%. 

At 7% it will be -|- more than at 6%. 

At 2% it will be of that at 6%. 

At 4% it will be 1 less than that at 6%. 

At 8% it will be J more than that at 6%. 

At 9% it will be more than that at 6%. 

At 10% it will be ^ of that at 6%. 

120. Find the interest on $250 for 1 year, 3 months 
and 20 days at 7% pr. annum. 

$2.50 Intercut for 00 days at 6%. 
? I, = Number of GO days. 

125 " 
1750 

20 <Zoya. 



a 6%, 
3.263=/ttteresl at 1%. 

$22.8S=Interest at 7%. 

or $22.85. 

121. 708.18 for 6 yrs 1 mo 6 ds 9% pr. an.=? 

122. $1000.00 for 4 yrs 2 mos 4 ds @ 7^ pr. au.r=? 
Answers: $438.10, $292.44. 

123. $340 for 2 yrs 2 mos 20 ds @ 2J% pr. an.=? 

124. $000 for 3 yrs 4 mos 15 ds @ 6%% pr. an.= ? 

125. $850 for 1 yr 2 mos 12 ds at \% pr. an.=? 
Total, $237.22. 

Find the interest of 

126. $617.18 for 3 mos 18 ds @ 15% pr. an. 

127. $460.74 for 2 mos 5 ds @ 18% pr. an. 

128. $765.12 for 8 mos 16 ds @ 20% pr. an. 
Total, $151.55. 



SIMPLE INTEREST. 151 

Find the interest on the following at 10% per annum: 

129. $710 for 92 days, 133. $496 for 91 days. 

130. $19^8 for 27 days. 134. $671 for 80 days. 

131. $8889 for 128 days. 135. $100 for 104 days. 

132. $75 for 117 days. 136. $269 for 73 days. 

Total, $351.47. Total, $36.91. 

163. It is customary for bankers to lend money, and 
discount by the month instead of the year. This percent- 
age is easily converted into 6% interest, and the work 
performed with as much ease as before. 

1 % per month is 12% per year, or 2 times 6%. 
l-J-% per month is 18% per year, or SHimes 6%. 

2 % per month is 24% per year, or 4 times 6%. 

Find the interest on the following: 

137. $65 for 80 days @ 2 % per month. 

138. 840 for 33 days @ \\% per month. 

139. $190 for 63 days @ 2 % per month. 

140. $700 for 93 days @ 3 % per month. 
Total, $77.20. 

Find the amount of the following 

141. $710 for 36 days @ l-J-% per month. 

142. $216 for 45 days @ 2 % per month. 

143. $1800 for 57 days @ lJ-% per month. 

144. $560 for 14 days @ 6J% per month. 
Total, $3367.89. 

When computing interest, the ingenious student will 
contrive many ways for abbreviating his work. Some- 
times he will take advantage of the aliquots of 100; at 
other times he will transpose the terms, and consider the 
days as dollars and the dollars as days, or he will reduce 
the rate mentally to 6%, if.it is some other rate, and thus 
simplify as well as abbreviate. For instance, in the ques- 



152 NELSON'S COMMON-SCHOOL ARITHMETIC. 

tion 138, lie might consider the $40, 120, and transposing 
the terms, which could be done instantly, he would simply 
have to multiply the 33 by 2, making 66c, the answer. 

It will materially abridge the operation and expedite the 
labor, if the learner will observe to avoid the use of all 
lines, figures or marks that arc not absolutely necessary. 
As, for instance, when using aliquot parts, to write only 
the results of division, as shown in the following example: 

145. Interest of $321 for 2 years, 1 month 3.21 

and 22 days at 10% per annum. 40 125 

EXPLANATION. Mentally it is found that there are 12 J 1.07 
60 days in 2 years and 1 month, to multiply by which we 107 

divide by 8. The division by 6 and the multiplication ji 0^9 

by 10 were performed simultaneously, giving $08.836 

or $08.84 as the answer. 68.836 

In Bank Discount, the learner will find numerous ques- 
tions upon which to exercise his ingenuity, arid the judi- 
cious teacher will encourage him. 

The method of finding True Discount, and other more dif- 
ficult calculations, will be found at the end of Banking. 

PARTIAL PAYMENTS, OR PAYMENTS BY 
INSTALLMENTS. 

164. Notes, bonds, etc., drawing interest, are sometimes 
paid by installments, and the amounts thus paid indorsed 
on them. The legal rule for computing interest on install- 
ments may be expressed thus: 

Apply the payment to the discharge of the interest, and 
if there is a remainder, subtract it from the debt. When 
the payment is less than the interest due, it is not applied 
to the discharge of the interest or debt, but is indorsed 
on the note until the installments exceed the interest. 
The sum of the installments is then taken from the amount 
due, and interest computed on the remainder as before. 






SIMPLE INTEREST. 153 

1. $576. CINCINNATI, Oct. 9, 1857. 

On demand, I promise to pay Robert Ingles, or 
order, Five hundred and seventy-six dollars, with interest. 
Value received. SAMUEL DUNNING. 

On the note are the following indorsements : 
Rec:d, Dec. 16, 1857, $100. 

" Pel. 28, 1858, 3. 

" July 27, 1858, 150. 
Required the amount due September 3, 1858. 

Yrs. Mos. J)s. 

From 1857 12 16 
Take 1857 10 9 
Difference, 2 7 or 67 days. 

$576.00 ^4 wowntf of note. 

6A3=Intercst on $576 for 67 days. 

$582.43= Total amount due. 

to be subtracted. 



2.43= Balance due. 



The second payment is less than the interest due and 
no calculation is required. 

From December 16, 1857, to July 27, 1858, is 7 months 
11 days. 

$482A3=:Balance due. 

17 .! Interest for 7 mos. 11 days. 



$500.18 .^moimtf due. 
153.QQ= Amount of payments, 

$347 '.18= Balance due. 
From July 27 to September 3, is 38 days. 



2.W=Literest for 38 days. 

due September 3, 1858. 



154 NELSON'S COMMON-SCHOOL ARITHMETIC. 

2. $650. BOSTON, June 3, 1848." 

For value received, I promise to pay, on demand, 
to H. Crooks, or order, Six hundred and fifty dollars, 
with interest at 6% per annum. J. F. DAVIS. 

Indorsements. 

Jan. 6, 1850, $95. 
Oct. 13, 1850, 350. 
June 3, 1855, 12. 
Sept. 7, 1857, paid the balance; how much was it? 

Aris. $405.92. 

3. On a note drawn September 23, 1857, for $650, with 
legal interest, there are the following indorsements : 

Oct. 4, $100. 
Nov. 3, 2. 
Dec. 11), 210. 
April 3, 1858, the balance; how much was it? 

Am. $354,32. 

4. On a. note drawn October 3, 1856, for $1000, with 
interest, are the following indorsements: 

Nov. 30, $50. 

Jan. 6, 1857, 100. 
Feb. 9, " 5. 

June 6, " 190. 
Feb. 3, 1859, the balance; what was the amount? 



XVIII. BANKING. 

164. Banking is the business of dealing in money. 
Banking houses borrow and lend money, receive money 
for safe keeping, and exchange the money of one country 
for that of another. There are National Banks, Public 
and Private Banks, Banks of Deposit, Banks of Issue or 
Circulation and Banks of Discount. 



BANKING. 155 

A National Bank is one which issues notes secured by 
Donds of the United States, deposited with the United 
States Treasurer, and is doing business under the authority 
of the General Government. 

A Public Bank is one that is owned by a joint stock 
company, who commit its management to some of their 
number chosen for that purpose. These persons are called 
President and Directors. 

A Private Bank is one which is owned by one or more 
individuals, who attend to its business personally. 

Banks of Deposit receive the ready money of merchants 
and others for safe keeping. Banks of deposit also loan 
money on interest and some pay interest on deposits. 

Banks of Issue or Circulation manufacture and issue 
paper money, called bank-notes or bills. Many of these 
batiks also receive money on deposit and do a discount 
business. 

Banks of Discount lend money on interest, when suita- 
ble security is given. 

The security required by banking institutions when 
loaning money, is a note or notes from the borrower, with 
the names of one or two responsible persons written on 
the back, in such a manner as to bind them for payment, 
should the drawer of the note fail to pay it at the proper 
time. The persons who sign their names thus are called 
indorsers, the writing the indorsement, and the person who 
writes the note and signs his name, the maker or drawer. 

The notes given by borrowers may be their own* or 

* Notes are always known by the names of the makers. We 
Bpeak of A B 1 s note, though it is in our possession. 

To hold an indorser, he must be notified of the maturity of the 
note at furthest on the next business day, if he resides in the same 
town or city ; otherwise, a notice should be mailed to him within 
the same time. 



156 NELSON'S COMMON-SCHOOL ARITHMETIC. 

those of other persons. If those of others, the borrower's 
indorsement alone is all that is usually asked. 

The person to whom the note is made payable is called 
the payee, and the person possessing it, the holder. By 
the face of the note is meant the side containing the prom- 
ise; also the amount for which it is drawn. 

A negotiable note is one that can be transferred. To 
be negotiable, it should be drawn "to the order' 1 of the 
payee, or to him " or order" or "to bearer." In the first 
two cases, to make it negotiable, the payee would have to 
write his name on the back. To the third class belong 
bank-notes, whic^ may be passed by any holder without 
indorsement. 

The phrase "value received" is not considered essen- 
tial to a note, though it is well to insert it. 

Days of Grace are three days allowed to the maker of 
a note beyond the stated time of payment. A note that 
is drawn payable 90 days after date, is not legally due 
until the 93d day. 

165. MATURITY OF NOTES. 

Notes falling due on Sunday or a national holiday are 
payable the day before. 

A note dated the 28th, 29th, 30th or 31st of Jan- 
uary, at one month, falls due on the last day of February 
without, or on the 3d day of March with grace ; and a 
note dated the last day of February, at one month, falls 
due on the 28th of March, if not a leap-year ; otherwise, 
on the 29th of March, plus the days of grace. Hence, a 
note drawn by the month, and dated on the last day of the 
month, falls due on the same day of the month, if the 
latter month have a corresponding day; otherwise, it falls 
due on the last day of the month. Thus, a note dated 



BANKING. 157 

November 30th, at two months, without grace, would fall 
due on the 30th of January ; at three months, oil the 28th 
or last day of February. 

166. REMARKS ON NOTES. 

1. A note need not be dated at the place where it is 
drawn, and can be made payable at any particular place 
the parties may agree upon; but to hold an indorser, de- 
mand must be made at the particular place specified. A 
note is also good if dated on Sunday. 

2. When giving a note, the maker ought to fix the place 
Df payment say his bank or place of deposit if the payee 
be agreeable. 

3. In New York and other States, notes draw seven per 
cent, interest. Notes made payable in those States, accord- 
ingly, draw that rate of interest after maturity. 

4. If made by more than one person, it is called a joint 
note, or joint and several. It is a joint note only, unless 
words are used to indicate individual responsibility, as 
"We jointly and severally," etc. 

5. It is customary for merchants to deposit their notes 
in bank for collection, the bank, by this means, undertak- 
ing to see them paid, or using means to hold the indorsers. 

6. A note drawn payable to the maker, and by him in- 
dorsed in blank, can be transferred negotiated without 
liability to the holder. 

7. When taking up a note paying it the drawer 
ought to require the indorsement of the holder, deface his 
own signature and file the note away, as it may subserve 
the purpose of a receipt at some future time. 

8. A note drawn under the seal of the maker is called 
a bond. 

Indorsements are of various kinds, and, like the note 
itself, they require no stated form of words. 



158 NELSON'S COMMON-SCHOOL ARITHMETIC. 

There are Blank, Full, Special and Restrictive indorse- 
ments. 

A Blank Indorsement is made by the payee or holder 
writing his name on the back. 

A Full Indorsement, or, as it is called, an indorsement 
in full, is made by writing such a transfer as "Pay to the 
order of A," or words to that effect, and signing the name. 

A Special Indorsement is one made to suit a particular 
case, as when the indorser wishes to free himself of re- 
sponsibility should the maker fail to pay the note. "Pay 
the the contents to B, or order, without recourse on me." 
"Payee." 

A Restrictive Indorsement restrains the pa} r ment of the 
note to the party to whom it is indorsed, as "Pay the con- 
tents to C only." This kind of indorsement does not limit 
the payment when made by another person than the payee 
of the note. 

The payee is called the first indorser, if he transfers the 
note, and the party to whom he transfers it, the indorsee. 
Should the latter again transfer it, he would be called the 
second indorser and the party to whom it was transferred, 
the second indorsee. 

Indorsements ought to be written across the back, with 
the left end up, as shown on page 160. 

167. PtEMARKS ON INDORSING. 1. A holder or indorsee 
can write over a blank signature a full or restrictive in- 
dorsement. 

2. A holder may cancel all indorsements (signatures) 
but the first, and may cancel a full indorsement, except 
the signature of the first. 

3. The object of a full indorsement is to prevent its 
transfer without the signature of the holder. 

4. A note may be transferred after it is due, but the 
indorser is not then liable. 



BANKING. 159 

5. The payee or his agent must make the first transfer. 

6. Any partner of a firm may indorse for all. After 
maturity, it ought to be done severally. 

7. Agents may indorse for principals thus: 

A B, Principal, 

By C D, Agent. 

8. An indorsement cancelled by mistake does not dis- 
charge the indorscr. 

FORMS OF PROMISSORY NOTES. 

The following are given as examples for the learner to 
to copy. The printed blanks differ from these, and are 
easily filled. 

A NEGOTIABLE NOTE. 

CLEVELAND, Sept. 30, 1866. 

Thirty days after date, I promise to pay to the order 
of J. C. Hutsiupiller Five hundred sixty T 3 ^j- dollars, at 
the First National Bank, this city, value received. 

$560^. E. R. FELTON. 

A NON-NEGOTIABLE NOTE. 

MILWAUKEE, Apr. 4, 1867. 

Ninety days after date, I promise to pay to Gr. W. Nel- 
son One thousand dollars, value received. 
$1000. R. C. SPENCER. 

A JOINT AND SEVERAL NOTE WITH INTEREST. 

CINCINNATI, Aug. 9, 1867. 

On demand, six months after date, we severally and 
jointly promise to pay to the order of William Otte Six 
thousand dollars, value received, with interest at six per 
cent, per annum. GEORGE F. SANDS, 

W. J. BREED. 



160 NELSON'S COMMON-SCHOOL ARITHMETIC. 
FORM OF INDORSEMENT. 




When writing the following notes, the learner ought to 
use cap paper, so that the notes be long enough for 
filing, and leave a margin above and below of not less 
than one and a half lines. 



HOME EXERCISES IN DRAWING NOTES. 

Draw notes from the following data: 

1. Date, Aug. 18, 1868. 

Place, Louisville. 

Maker, Yourself. 

Time, On demand. 

Face, $678.14. 

Payee, S. E. Peyton. 

3. Date, April 9, 1870. 
Place, Pittsburgh. . 
Drawer, S. S. Sargent. 
Payee, Yourself. 
Face, $168.75. 
Time, 3 months. 



2. 


Date, 


Nov. 6, 1869. 




Place, 


Cincinnati. 




Drawer, 


Yourself. 




Payee, 


James Moore. 




Fnce, 


$375.50. 




Time, 


90 days. 


4. 


Date, 


Dec. 3, 1869. 




Place, 


Louisville. 




Time, 


On demand. 




Drawer, 


E. P. Bowers, 




Payee, 






Indorser, 


R. Johnson. 




Face, 


$6745.15. 



BANKING. 



161 



5. Date, Oct. 19, 1860. 6. Date, 

Place, Wheeling, Va. 
Drawer, Archibald Warren. 
Payee, Yourself. 

Indorser, 

Face, $1364. 
Time, 6 months. 
Indorsee, Saml. Adams. 



Nov. 29, 1867. 



Date, 

Place, 

Time, 

Face, 

Drawer, 

Payee, 

1st Indorser, , 

Indorsee, Yourself. 



April 3, 1869. 
New York. 
4 mouths. 
$1408.75. 
James Todd. 
Robt. Emmet. 



9. Place, 
Date, 
Time, 
Maker, 

Payee, 

1st Indorser, James Moore. 
2d Indorser, Yourself. 
Face, 



Mobile. 10. 

April 3, 1867. 
One day afterdate. 
S. 0. Miner. 



11. 



Place, Boston, Mass. 

Date, July 10, 1870. 

Time, 30 days. 

1st Indorser, Robt. Penn, 
1st Indorsee, James Moon. 



Place, New Orleans. 

Drawer, E. M. Small. 
Payee, H, C. Piner. 

1st Indorser, 

1st Indorsee, Edward Epply. 

2d Indorser, 

Face, $138. 

Date, Jan. 1, 1867. 
Place, Chicago. 
Time, 60 days. 
Drawer, R. E. Baird. 
Payee, J. E. Piner. 

Indorser, 

Indorsee, J. J. Wood. 
Face, $1637.00. 

Place, San Francisco. 
Date, 

Time, 4 months. 
Maker, Henry Adams. 
Indorsee, Thomas Orr. 
Payee, Henry Adams. 

Indorser, 

Face, $450. 

Maker, S. Poor. 

Payee, 

2d Indorser, ..* 

2d Indorsee, Wm. A. Miller. 
Face, $396.57. 



Draw the notes necessary for the following transac- 
tions : 

12. Take H. P. Spike's obligation, at 1 day after date, 
for price of a house worth $1695.* 



*To make this a bond, insert the word "seal" after the name 
14 



162 NELSON'S COMMON-SCHOOL ARITHMETIC. 

13. Pay S. 0. Mooney 167.87 with your note at 30 
days, bearing interest. 

14. Lend Henry Shotwell 1687, on his note, in such 
a way as to enable you to collect the money at any time. 

15. Jno. Emerson's note, our favor, indorsed to G. 
Young, for 675 at 90 days, was protested for non-pay- 
ment on January 3, 1866. What was its form? 

16. Renew your note of $190, and take 3 months' time, 
allowing interest for the time the new note has to run. 

17. Pay Juo. Sallust 367, on acct., with a note at 90 
days. 

18. Indorse C D's note of Sept, 3, I860, for 6600 at 90 
days (your favor), in such a way that you will not be 
held responsible for payment. 

19. Give W. A. Moore your note at 3 months, and 
draw it in such a way as will prevent him from negotiat- 
ing it. 

20. Exchange notes with your neighbor, each drawing 
at 90 days, and indorse. 

21. Draw a copy of your note for 675, favor of H. 
Corncob, at 3 months, upon which you had to pay $2.47 
interest on Sept 19, at 6%. 

22. Pay Saml. Saul Pickwick the balance of S678.50, 
you owe him, by giving note in favor of Sam. Weller at 
60 ds. 

23. Take the note of J. Morgan Henry, Madison, Ind., 
for $360, at 3 months, and secure yourself against the 
statute which allows " relief," etc. 

(After the words "value received," insert "without any relief 
whatever from valuation or appraisement laws." For form of 
note, see page 202.) 

24. Indorse the receipt of 167.87, amount paid you 
on E. C. Johnston's note, dated Aug. 9, 1866, and drawn 
for 974.35 at 6 months, your favor. 



BANKING, 163 



CHECKS DRAFTS BILLS OF EXCHANGE. 

A Check is a written order on a bank or its cashier for 
the payment of money in its possession belonging to the 
party making it. A Draft* is similar to a check, but is 
written more formally and may be drawn upon any other 
than a banker; and a Bill of Exchange is a check or draft 
used for transmitting money to distant places. 

A check, like a promissory note, may be drawn to 
bearer, to order, or be made payable to a particular per- 
son. When given to strangers, or for large sums of money, 
they should be drawn to order, so that those receiving 
them would have to indorse them before payment. In 
this way, they may also be used to subserve the purposes 
of receipts. 

Checks may be antedated or post-dated. In the former 
case, they are payable on presentation, if there are funds 
of the drawer to meet them; in the latter, they will not 
be payable till the date arrives. Should a bank refuse to 
pay a check, the party to be sued is the maker, or trans- 
ferrer, if it has been transferred. 

If a bank pay a forged check, the bank, and not the 
person whose signature is forged, has to sustain the loss, 
though the forgery may be so well executed that it caa 
not be detected by ordinary inspection. 

When a check is so carelessly drawn that an alteration 
may be easily made, the loss arising must be borne by 
the drawer. For instance, the amount to be paid should 
commence at the extreme left of the line, and the part 
left unfilled be written with a curved line or other mark 
to prevent additions. 

*For forms of draft, see Exchange. 



164 NELSON'S COMMON-SCHOOL ARITHMETIC. 
FORM OF CHECK. 




OUDERS DUE-BILLS. 

A mercantile Order is a request for the delivery of 
goods or money. It differs from a draft in being more 
simple in its form, and its being usually drawn for the 
payment of goods instead of money. The following is the 
common form: 

CINCINNATI, Jarfy 1, 1867. 
MESSRS. JOHN SHILLITO & Co. 

Will please let Samuel Steele, or bearer, have goods to 
the amount of One hundred dollars, and charge to my ac- 
count. 

$100^. GEORGE KIDD. 

To be more specific, the kind of goods and the price, 
wholesale or retail, might be inserted. 

A Due-bill is simply an acknowledgement of a debt. 
It is usually drawn for a small sum and to settle an ac- 
count. 

Due-bills are considered to draw interest from date, 
though it is seldom exacted. 






BANKING. 165 

FORM OF A DUE-BILL. 

INDIANAPOLIS, Aug. 3, 1868. 

Due Henry Ellsworth, or order, Fifteen dollars. Value 
received. 
$15_QLQL > HENRY SPENCER. 

RECEIPTS. 

It may not be considered improper here to introduce a 
few forms of receipts, as they are essentially connected 
with the preceding subjects. 

A Receipt should specify what it was given for, whether 
money, goods, note, etc., the amount for which it was 
given and the date, and the amount should be in writing. 
Receipts for sums over twenty dollars should have a stamp. 

Receipts may be made on bills, notes, etc., or given 
separately, and need not be of any particular form. 

RECEIPT ON ACCOUNT. 

CINCINNATI, Jan. 1, 1868. 

Received of Mr. John Cummins, One hundred and 
twenty-five dollars and 23 cents on account. 

-. JAMES MORGAN. 



RECEIPT FOR MONEY ON NOTE. 

PITTSBURG, Apr. 3, 1867. 

Received of Alex. Cowley, One thousand dollars, to be 
credited on his note, my favor, dated Jan. 3, 1867, for 
six thousand dollars. 
$1000. W. ALLAN MILLER.* 



*A payment on a note should be receipted (indorsed) on the back 
of the same, and a statement made that a receipt was given. 



166 NELSON'S COMMON-SCHOOL ARITHMETIC. 

RECEIPT IN FULL. 

Received, Cincinnati, June 3, 1SG8, of Timothy Hay, 
Thirty -five -,YV dollars, in full to date. 
$35.25. THOMAS E. YOUNGMAX. 

RECEIPT FOR RENT ON ACCOUNT. 

BOSTON, Feb. 27, 1867. 

Received of Mr. Henry G. Judkins Sixty-eight dollars, 
on account of rent for No. 6 Long St. 
$68.00. EDW. FABER. 

RECEIPT FOR A NOTE. 

PHILADELPHIA, Jariy 30, 1869. 

Received of Mr. James Thompson, his note, my favor, 
at ninety days, for Five hundred dollars to balance acct. 
$500. JNO. F. GREEN, JR. 

RECEIPT FOR MONEY IN ADVANCE. 

COLUMBUS, Apr. 3, 1869. 

Received of J. Q. A. Miller, Forty dollars, in advance, 
for live pork, to he delivered to him on or before October 
1 1869, at 7 cents a pound. 
$40. JAS. HOLLANDER, SEN. 

RECEIPT FOR MONEY RECEIVED ON ACCOUNT OF ANOTHER. 

CINCINNATI, Oct. 10, 1866. 

Received from H. D. Brown, on the account II. W. 
Sou they, Sixty-nine y 8 ^ dollars, in full of acct to 1st inst. 
$69^. Y. W. COOK, JR. 

RECEIPT FOR RENT IN FULL. 

NEW YORK, Sept. 12, 1866. 

Received from R. Quinn, Twenty-five dollars, in full for 
rent of house, No. 96 Chestnut St., to 9th iust. 
$25. JAS. M. HALL. 



BANKING. 167 

RECEIPT FOR MONEY RECEIVED BY A CLERK. 

NEW YORK, Aug. 3, 1870. 

Received of II. Simon One hundred dollars on acct. 
. JAMES MOORE. 

Per JNO. WOOD, Clk. 



HOME EXERCISES IN DRAWING RECEIPTS. 

1. Receipt to John Roberts for $100 on account. 

2. Give H. C. Parker a receipt for $50 in full of ac- 
count. 

3. Draw a receipt for $250 T 5 ,?g- in favor of C. C. Martin, 
for his note of this date in settlement of account, and 
draw a copy of the note, 

4. Receipt to Mrs. T. H. Henshaw for $365, in part 
for rent of house, 1968 Vine Street. 

5. Give your teacher a receipted bill for 257 barrels of 
flour at $12.67 per barrel; drayage, $8.50. 

6. Let William Eduiuridson have your receipt for $67, 
on his note, of the 3d of last month, at 6 months, your 
favor; and draw a copy of the note, showing a receipt to 
be made on it. Face of note, $936. 

7. M*ke out to your teacher, this date, the bill on page 
111, and take a note in settlement. 

8. Receipt for $150 cash on the first bill, page 112, 
making it to William Nelson. 

9. As clerk of II. J. Estcourt, make out the first bill 
on page 113 to your teacher, allowing him a discount of 
5% for cash. 

10. Give your due-bill and an order on your grocer, to 
teacher, for tuition one session of 5 months, amounting to 
$85. Make the order for a balance of $15.75 in grocer's 
hands, and draw all the papers. 



168 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



TIME TABLE 

FOR COMPUTING INTEREST AND AVERAGE. 

Number of days from Is/ of Jan-nary to any other day of the year. In leap-years, add I 
to the days after 26th of February. 



a 



o 

i 


j.Tiinmry .... 


I jFobrujir; 



*i 


* 

T3 


g 

S 
X 


-l 
S3 

3 

0> 


,Day of Mo. 


^ 

e^ 
5T 


I August 


cc 

4 

3 
1 


I 

n 


1 1 November 


i December. 


I! 

Day of Mo. 

i 














: 1 : 
























1 





:51 


59 


90:120 


151 


1 


181 


212j243|273 


304 


334 


1 


2 


1 


32 


60 


91 


121 


152 


2 


182 


213 


244i274 


305 


335 


2 


3 


2 


33 


61 


92 


122 


153 


3 


183 


214 


245 


275 


306 


336 


3 


4 


3 


34 


62 


93 


123 


154 


4 


184 


215 


246 


276 


307 


337 


4 


5 


4 


35 


63 


94 


124 


155 


5 


185 


216 


247 


277 


308 


338 


5 


6 


5 


36 


64 


95 


125 


156 


6 


186 


217 


248 


278 


309 


339 


6 


7 


C 


37 


65 


96 


126 


157 


7 


187 


218 


249 


279 


310 


340 


7 


8 


7 


38 


86 


97 


127 


158 


8 


188 


219 


250 


280 


311 


341 


8 


9 


8 


3! 


67 


98 


128 


159 


9 


189 


220 


251 


281 


312 


342 


9 


10 


9 


40 


68 


99 


129 


160 


10 


190 


221 


252 


282 


313 


343 


10 


11 


10 


41 


69 


100 


130 


161 


11 


191 


222 


253 


283 


314 


344 


11 


12 


11 


42 


70 101 


131 


162 


12 


192 


223 


254 


284 


315 


345 


12 


13 
14 


12 
IB 


43 
44 


71 102132 
721031133 


163 
164 


13 
14 


193 
194 


224 
225 


255 
256 


285 
286 


316 
317 


341) 
347 


13 
14 


15 


14 


15 


73 


104 


134 


165 


15 


195 


226 


257 


287 


318 


348 


15 


16 


15 


46 


74 


105 


135 


166 


16 


196 


227 


258 


288 


319 


349 


16 


17 


1(5 


47 


75 


106 


136 


167 


17 


197 


228 


259 


289 


320 


350 


17 


18 


17 


48 


76 


107 


137 


168 


18 


198 


229 


260 


290 


321 


351 


18 


19 


18 


49 


77 


108 


138 


169 


19 


199 


230 


261 


291 


322 


352 


19 


20 


19 


50 78 


109 


139 


170 


20 


200 


231 


262 


292 


323 


553 


20 


21 


20 


51 


79 


110 


140 


171 


21 


201 


232 


263 


293 


324 


354 


21 


22 


21 


52 


80 


111 


141 


172 


22 


202 


233 


264 


294 


325 


355 


22 


23 


22 


53 


81 


112 1421173 


23 


203 


234 


265 


295 


326 


356 


23 


24 


23 


54 


82 


113|143 


174 


24 


204 


235 


266 


296 


327 


557 


24 


25 


24 


55 


83 


114 


144 


175 


25 


205 


236 


267 


297 


328 


-558 


25 


26 


25 


56 


84 


115 


145 


176 


26 


206 


237 


2681298 


329 


359 


26 


27 


26 


f)7 


85 


116 


146 


177 


27 


207 


238 


269 


299 


330 


360 


27 


28 


27 


58 


86 


117 


147 


178 


28 


208 


2391270 


3001331 


561 


28 


29 


28 




87 


118 


148 


179 


29 


2091240 


271 301 


332 


362 


29 


30 


29 




88 


119 


149 


180 


30 


210 


241 


272 


302 


333 


363 


30 


31 


30 




89 




150 




31 


211 


242 




303 




364 


31 



BANKING. 



169 



TIME TABLE 

FOR COMPUTING INTEREST AND AVERAGE. 

Number of days from 1st of July to any other day of the year. In leap-years, add 1 to 
the days after 28//t of February. 



b 

p 

s. 

o 




fT 


August , 


^September .. 


I ^ 
i g. 

o 
^ 


1 November ., 


! December... 


Day of Mo... 


[January 

I 


i February.... 


|March 


> 

V 




" | 
<*$ 


-| 

Bl 






'Day of Mo... 

i 


1 





31 


62 


92 


123 


153 


1 


1841215 243 


274 


304 


335 


1 


2 


1 


32 


63 


93 


124 


154 


2 


185 


216 


244 


275 


305 


336 


2 


3 


2 


33 


64 


94 


125 


155 


3 


186 


217 


245 


276 


306 


337 


3 


4 


*j 


34 


65 


95 


126 


156 


4 


187 


218 


246 


277 


307 


338 


4 


5 


4 


35 


66 


96 


127 


157 


5 


188 


219 


247 


278 


308 


339 


5 





5 


36 


67 


97 


128 


158 


6 


189 


220 


248 


279 


309 


340 


6 


7 


6 


37 


68 


98 


129 


159 


7 


190 


221 


249 


280 


310 


341 


7 


8 


7 


38 


69 


99 


130 


160 


8 


191 


222 


250 


281 


311 


342 


8 


9 


8 


39 


70 


100 


131 


161 


9 


192 


223 


251 


282 


312 


343 


9 


10 


9 


40 


71 


101 


132 


162 


10 


193 


224 


252 


283 


313 


344 


10 


11 


10 


41 


72 


102 


133 


163 


11 


194 


225 


253 


284 


314 


345 


11 


12 


11 


42 


73 


103 


134 


164 


12 


195 


226 


254 


285 


315 


346 


12 


13 


12 


43 


74 


104 


135 


165 


13 


196 


227 


255 


286 


316 


347 


13 


14 


13 


44 


75 


105 


136 


166 


14 


1971228 


256 


287 


317 


348 


14 


15 


14 


45 


76 


106 


137 


167 


15 


198 


229 


257 


288 


318 


349 


15 


16 


15 


46 


77 


107 


138 


168 


16 


199 


230 


258 


289 


319 


J50 


16 


17 


16 


fc7 


78 


108 


139 


169 


17 


200 


231 


259 


290 


320 


351 


17 


18 


17 


48 


79 


109 


140 


170 


18 


201 


232 


260 


291 


321 


352 


18 


19 


18 


49 


80 


110 


141 


171 


19 


202 


233 


261 


292 


322 


353 


19 


20 


19 


50 


81 


111 


142 


172 


20 


203 


234 


262 


293 


323 


i&4 


20 


21 


20 


5182J112 


143 


173 


21 


204 


235 


263 


294 


324 


355 


21 


22 


21 


52 


831113 


144 


174 


22 


205 


236 


264 


295 


325 


356 


22 


23 


22 


53 


84lll4 


145 


175 


23 


206 


237 


265 


296 


326 


}57 


23 


24 


23 


54 


85 115 


146 


176 


24 


207 


238 


266 


297 


327 


358 


24 


25 


24 55 


86116 


147 


177 


25 


208 


239 


267 


298 


328 


359 


25 


26 


25 


5(5 


871117 


148 


178 


26 


209 


240 


268 


299 


*29 


J60j 


26 


27 


26 


57 


88 


118 


149 


179 


27 


210 


241 


269 


300 


330 


'361 


27 


28 


27 


58 


89 


119 


150 


180 


28 


211 


242 


2701301 


331 


362 


28 


29 


28 


59 


90 


120 


151 


181 


29 


212 




271!302 


332 


'363 


29 


30 


29 


60 


91 


121 


152 


182 


30 


213 




272 


303 


333 


364 


30 


31 


30 


61 




122 




183 


31 


214 




273 




334 


j SI 



15 



170 NELSON'S COMMON-SCHOOL ARITHMETIC 

USE OF THE PRECEDING TABLES. 

1. From ~[st of July to 9//i of June, how many days? 
Opposite 9, in the last column, is 342, the answer. 

2. From 2d of October to 17th November, liow many days? 
Opposite 2, in the middle column, is 93, in October 

column, and opposite 17 is 139, in the November column. 
The difference is 46. 

These tables are specially adapted for averaging ac- 
counts, taking 1st of January and 1st of July as dates 
from which to reckon. 

DISCOUNTING NOTES. 

168. Discounting notes consists in buying them at less 
than their nominal value, or the amount for which they 
are drawn. The difference between the nominal value and 
the price paid is called discount. 

169. Bankers prefer lending money on short time, and 
by the day, instead of by the month. Notes are usually 
drawn for 30, 60 or 90 days; and interest is always 
charged on the days of grace. 

170. There are two kinds of discount: True Discount, 
which is the interest paid in advance on the present value 
of a note, and Bank Discount, which is interest paid in 
advance on the face of the note. The latter resembles 
compound interest, as it is interest on both interest and 
principal.* 

When a note is discounted in bank, the interest of the 
note for the time it has to run, and at the banker's rates, 
is deducted from the sum called for by the note. The bal- 

*The present worth of a note drawn for $100, payable in a year 
at 6 per cent., is $94.84, and the interest is $5.56; that is, the prin- 
cipal and interest together are equal to $100, or the face of the note; 
BO when a banker discounts from the face of a note, he discounts ou 
both principal and interest. 



BANKING. 171 

ance is called the proceeds. This species of discount is 
therefore reckoned in the same way as interest. Bankers 
reckon interest on every day intervening between the day 
of discount and that of maturity, including the latter. 

171. When a note is drawn by days, subtract the ex- 
pired term from the number of days for which it is 
drawn, plus the days of grace ; but when drawn by the 
month, first find the day of maturity and reckon the 
whole number of days from the day of discount to that 
date. 

1. Discounted on the day of date, how much discount 
jhould be deducted from a note of $500 at 90 days?* 

$5. 00= Interest for GO days. 
2.50== " " 30 " 
.25^ 3 " (grace.) 

Arts. $7.75 

2. $1500. COLUMBUS, Jan. 8, 1859. 

Sixty days after date, I promise to pay Messrs. 
M'Ewen and Banfill One thousand five hundred dollars, 
value received. GEO. K. TENNEY. 

Required, the discount at 6% per an. Ans. $15.75. 

3. $3500. WHEELING, Oct. 3, 1858. 

Ninety days after date, I promise to pay John 
M'Culloch, or order, at First National Bank, three thou- 
sand five hundred dollars, value received. 

MILO G. DODDS. 
Required the proceeds at 6% per an. Ans. $3445.75 

4. Find the proceeds of a note for $120 at GO days at 
\% P er month. 

5. Required the proceeds of a note dated Jan. 1, 18Q6 J 
and drawn for $575.75 at 90 days. 

*Wken the rate is riot named, six per cent, per annum is under- 
stood. 



172 NELSON'S COMMON-SCHOOL ARITHMETIC. 

6. What is the bank discount on a note of $450 for 60 
days at 2% per month?* 

7 to 12. Find the proceeds of the following: 

On $850 at 30 days at \\% per month; on $1678.25 
at 90 days and \\% per mouth; on $670 at 60 days and 
2% per month; on $1749.57 at 90 days, 1J% per month; 
on $688 at 90 days, Z\% per month; on $6784 at 60 
days, If % per month. 

Answers: $118.74, $566.82, $344.22, $18.90, $1668.19, 
$6534.69, 640.01, $1613.22, $6594.69, $641.86. 

Find the discount on the following: 

13. $1310.00 for 60 days @ 2 % per month. 

14. $746.87 for 90 days @ \\% per month. 

15. $219.56 for 30 days @ 1 % per month. 

16. $1867.25 for 20 days @ 1\% per month. 

17. $1367.00 for 15 days @ 3 % per month. 

Total, $152.57. 

18. A note drawn on February 13, 1866, for $900, at 
90 days, was discounted on March 23, at 2% per month; 
how much was paid to the borrower? Am. $867. 

19. What proceeds should be paid on a note of $346: 
at 90 days, drawn on November 3, and discounted on De- 
cember 7, at \\% per month? Ans. $335.79.' 

20. A note for $689, made September 9, payable in 60 
days, was discounted on October 5, at 2% per month; 
what was the discount? Ans. $16.99.i 

While the cents in the principal are rejected in comput- 
ing interest and discount, they are always reckoned when 
finding the amount or proceeds. 

*Such questions as these may be abbreviated by mentally in- 
creasing the days or principal in the ratio that the rate is to 6 per 
cent. In this case, reckon interest on $450X4 or $1800 at G per 
cent, per annum. 



BANKING. 173 

Required the discount on the following: 

Face of note. Date. Time. When disc'd. Rate of discount. 

21. $167.50 Jan. 3, 1869, 60 ds, Feb. 7, 2 % per mo. 

22. $9876.00 Feb. 7, 1869, 90 ds, Mar. 12, 2J% per mo. 

23. $789.00 Juu. 18, 1869, 30 ds, July 3, l\% per mo. 

24. $1897.00 Feb. 21, 1869, 90 ds, Apr. 1, l|% per mo. 

Total, $555.24. 
Find the proceeds of the following : 

25. $676.37 Apr. 3, 1869, 90 ds, May 9, 2 % per mo. 

26. $679.39 Mar. 9, 1869, 30 ds, Apr. 3, ty% per mo. 

27. $7168.00 June 13, 1869, 60 ds, July 9, \\% per mo. 

28. $816.37 Aug. 12, 1869, 30 ds, Sep. 6, Z\% per mo. 

Total, $9172.40. 

29. A note for $4378.35, dated February 1, 1867, at 
4 months, was discounted on May 16 at 10% per annum; 
[required the proceeds. 

SOLUTION. This note falls due June 4. From May 
16 to that date is 19 days, giving for discount on $4378, 
823.11. Proceeds, $4355.24. 

30. Required the proceeds of a note dated September 3, 
1867, for $396.87, at 3 mouths, and discounted on Octo- 
ber 5, at \\% per month. 

31. A note dated September 30, at 3 months, and drawn 
or $1367.56, was discounted on October 1, at 2% per 
month; what were the proceeds? 

32. Required the proceeds of a note dated December 
31, 1867, drawn for $5363.75, at 2 months, and discounted 
February 1, at \\% P er month? 

33. A note for $1000, dated Februray 28, 1866, at 2 
months, was discounted on March 20, at 1% per month; 
required the proceeds. 

Answers: $986.00, $384.56, $1282.75, $5296.70 $529.50, 



174 NELSON'S COMMON-SCHOOL ARITHMETIC. 
Find the proceeds of the following: 

Fnre of note. Date. Time. When clisc'd. Kate of discount. 

34. $2676.00, Jan. 9, 1869, 90 days, Feb. 1, \\% pr mo . 

35. 87187.00, Feb. 3, 1869, 60 days, Mar. 13, l%% pr mo. 

36. -$768.21, Mar. 6, 1869, 30 days, Apr. 3, 2 % pr mo. 

37. $314.00, Apr. 7, 1869, 90 days, May 15, 2 % pr mo. 

Total discount, $181.95. 

Ain't of note. Date. Time. When discounted. Rate of disc't. 

38. $6785, Dec. 6, 1868, 6 mo. Dec. 29, 2 % pr mo. 

39. $3748, Jan. 3, 1868, 5 mo. Feb. 3, 1868, 2% pr mo. 

40. $6983, Mar. 9, 1868, 4 mo. June 8, 1868, l\% pr mo. 

Total proceeds, $16272.70. 

Amount. "Date. Time. When disc'd. Rate of discount. 

41. $3784, May 6, 2 mo. July 3, 2 % per mo. 

42. $6987, Jun. 8, 3 mo. Aug. 27, \\% per mo. 

43. $7854, July 24, 4 mo. Sept. 17, 1 % per mo. 

Total proceeds, $18371.58. 



XIX. TRUE DISCOUNT. 

172. TRUE DISCOUNT is the difference between the pres- 
ent worth of a note and the amount for which it is drawn. 

The present worth of a note or bill due at a future time 
without interest, is such a sum as would, if put at inter- 
est for the same time and rate, amount to the debt; and 
the difference between this sum and the debt is the discount. 

Except in courts of justice, this kind of discount is sel- 
dom used, business men preferring bank discount for its 
simplicity. 

DISCOUNT ON INTEREST-BEARING NOTES. Bankers dis- 
count off the value of the note, including interest, at matur- 
ity ; while Real Estate Brokers discount off the face of the 
note, plus the accrued interest, at the time of discount. 



TRUE DISCOUNT, ETC. 175 

1. What is the true discount on a note of $700 for 90 
days at 6%? 

The amount of a dollar for 93 days is $1.0155, by 
which, if we divide $700, we will find the present worth. 
OPERATION. $1.0155)700.0000(689.315 
60930 

NOTE. The interest, on $1 for 90 clays is 90700 

0155. The present value of $1.0155, for 93 81240 

days is, therefore, $1, and, accordingly, the " qjrn (\ 

present value of $700 for 93 days is $700 di- 91395 
rided by $1.0155 or $089.31, and the discount 



$700 $689.31, or $10.09. 32050 

30465 

PROOF. The interest on $689.315, 93 days, 15850 

s $10.683, which, if added to the principal, 10155 

will give $699.999 or $700. 



56950 
50775 
The pupil can prove his calculations by interest. 

2. What is the true discount on a note of $575 for 90 
days at 6%? 

3. What is the true discount on a note of $137.09 for 
90 days at 6%? 

4. What will be the proceeds of a note of $1878.67 at 
90 days, true discount? 

5. A note for $485.44, payable in 30 days after date, is 
worth how much, true discount? 

173. To find the FACE OF A NOTE, when the proceeds, 
time and rate are given. 

6. Required the principal wlien the proceeds are $275.23, 
le time 93 days and the rate 2% per month. 

Interest on $1 for 93 days at 2% per months. 062. 

Proceeds of $1=$1.000 .062=.938. 

Since there are as many dollars in the principal as the 
proceeds of $1 is contained times in the proceeds given, 
S275.230-r-.938 will give the principal required, $293.42+. 



176 NELSON'S COMMON-SCHOOL ARITHMETIC. 

PROOF. Interest on $293.42 for 93 days at 2% per 
month=18.19-|-, which, subtracted from $293.42., leaves 
$275.23, the proceeds. 

7. The proceeds are $212.60, time 63 days, rate 1J% 
per numth ; required the principal.* 

8. What principal will realize $120 proceeds in 6 months 
at 10% per annum? 

9. The time is three months, rate 10% per an., pro- 
ceeds $168.97; what is the principal? 

10. The rate is 12% per annum, proceeds $693.75, time 
4 months; required the principal. 

174. To find the RATE PER CENT., when the principal, 
interest and time are given. 

11. The principal is $300, time 60 days, interest $5; 
required the rate. 

Interest on $300 for 60 days at 6%=r$3. At 1%=.50. 
It is obvious that the rate will be as great as the number 
of times 1% is contained in the interest given. Hence, 
$5.00-f-50=the rate, 10%. 

PROOF. Interest on $300 for 60 days at 10% =$5. 

12. The principal is $396.15, time 13 months 9 days, 
interest $26.34,3; required the rate. 

13. What is the rate per cent, on $144 for 5 days, when 
the interest is 24 cents? 

14. Eequired the rate on $250 for 60 days, when the; 
interest is $3.50. 

175. To find the TIME, when the principal, rate per cent, 
and interest are given. 

Grace being allowed only on notes and drafts, where 
neither is named, it is not reckoned. 

*The learner can prove liis work by computing interest on the 
principal found. 



GENERAL EXERCISES 177 

K-t 

15. The principal is $1440, rate 10% per annum, inter- 
est $37. 50; required the time. 

Interest on $1440 for 1 day at 10% 40 cents. 

Since there are as many days as the interest for 1 is 
contained times in the interest given, $37.50-^40=93^ or 
94 days. 

PROOF. Interest on $1440 for 94 days at 10% per 
annum=$37.60.* 

16. The principal is $1674, rate 2% per month, interest 
$59.87 ; required the time. 

17. In what time will a note for $600, at 6% per an- 
num, draw $27.50 interest? 

1.8. A note for $375 drew $21 interest at 6% per an- 
num; how long did it require to do it? 

19. A merchant wishes to know the time it will take 
a balance of $917.50 to make $60.80, with interest at 10%. 

GENERAL EXERCISES. 

1. What is the bank discount on a note of $375, drawn 
at 90 days, at \% per month? 

2. What amount of proceeds should I receive from a 
note of $796, drawn at 60 days, 2% per month? 

3. What is the present value of a note drawn for $600 
at 30 days? 

4. What amount of money should I receive on a note 
of $675, discounted at 35 days (having 35 days to run), 
\\% per month? 

The Teacher can increase these exercises to any extent, as the 
pupils have to furnish proof of their work; hence, only a few have 
been given under each artlale. 

* Interest is never reckoned on the fraction of a day, hence the 
difference. 



178 NELSON'S COMMON-SCHOOL ARITHMETIC. 

5. June 3, discounted my note of $350 at 10%, having 
30 days to run ; required the discount. 

6. February 6, 1858, had A. Seers's note of $500, dated 
20th December, 1857, discounted at \\% per month, time 
to run, 33 days; what were the proceeds? 

7. June 3, 1858, discounted my note of December 9, 
1857, drawn for $1000, at 12 months, 10%; what had I 
to pay? 

8. October 3, lifted my note of 17th February, drawn 
for $1000, at 60 days; what amount of money had I to pay? 

9. July 30, had M. Norton's note of $360, dated 23d 
July, at 90 days, discounted at 2% per month, what sum 
of money (Hd I receive? 

Answers: $596.72, $11.63, $762.57, $663.19, $1027.50, 
$2.92, $339.36, $946.67, $491.75. 



XX. COMPOUND INTEREST. 

176. IN Compound Interest, the interest is converted 
into principal, every quarter, half year or year. Capital 
is thus more rapidly increased than by simple interest. 

Any person acquainted with the principles of simple 
interest will readily understand how to compute this. 

When there is a settlement of accounts between the 
parties, after interest has become due and interest is 
charged in the settlement, interest may be allowed upon 
the balance found due by the settlement. So an agree- 
ment, after interest is due, to turn it into principal is valid. 
Where there is a contract between the parties for the pay- 
ment of interest annually, if not paid, simple interest may 
be allowed upon the interest from the time it is due.* 

*Swan. 



COMPOUND INTEREST. 179 

1. What is the compound interest on $1000 for 2^ 
years at 6%, payable semi-annually (half-yearly)? 

The interest of 1000 for 6 mos., $30.00 

Add the principal, 1000.00 

Amount for 6 mo*., 1030.00 

Interest on 1030 for 6 mos., 30.90 

Amount for 1 .year, $1060.90 

Interest on 1060.90 for 6 mos., 31.827 

Amount for 18 mos., $1092.727 

Interest on $1092.727 for 6 mos, 32.78181 

Amount for Z years, $1125.50881. 

Interest on $1125.50881 for 6 mos., 33.76526' 

Amount for 2 years 6 mos., $1159.27407 

By deducting the prittopa/, 1000.00 

We have the comp\l int. for 2J yrs., $159.27 
t 

2. What is the compound interest and amount of $672 

for 4 years, at 6% per annum? 

By computing interest on $1 for a number of years, it 
will be found that the second amount is equal to the 
square of the first ; the third amount to the cube of the 
first; the fourth amount to the fourth power of the first; 
each power corresponding to the number of years. Hence, 
to find the amount of any principal for any number of years, 
it is only necessary to multiply the principal by the amount 
of $1 for the time and rate. Taking the last example, 
1.064=the amount for 4 years, which, multiplied by 672= 
. amount required. 

3. Find the amount of $375 for 20 years, at 6% com- 
pound interest, reckoned annually. 

At 6 per cent., money will double itself in 11 years, 10 months 
and 21 days; at 5 per v,eut., in 14 years, 2 months and 15 days; at 
8 per cent., in 23 yeai'6, 5 months arid lOJ days. 



180 NELSON'S COMMON-SCHOOL ARITHMETIC. 

4. Required the compound interest on $600 for 5 years 
at 1%. 

Answers: $241.53, $827.68, $848.38, $176.38, $137.50. 



XXI. AVERAGE. 

177. AVERAGE signifies mean or medium. An average 
number or quantity is one which has an* intermediate value 
between two or more numbers or quantities. The average 
between 3, 4 and 5 is 4. 

178. The average may be found by dividing the sum 
of the quantities by their number. 

1. Find the average between 3, 4, 5 and 8. 

3 EXPLANATION. In these four numbers there are 20 

4 parts ; in each of 4 number there are as many equal parts 

5 as 4 is contained in 20, viz., 6. 
_8 

4)20 
5 

2. What is the average price of 5 cows, which cost, re- 
spectively, $50, $60, $70, $80 and $90? 

3. The wages of 9 hands in a factory are $5, $6, $7, 
$8, $10, $11, $12, $15 and $16, respectively; what is the 
average ? 

4. Required the average length of the following pieces 
of calico: 37 yds., 35 yds., 31 yds., 30 yds., 32 yds., 27 J 
yds., 291 ^1^ 3QJ yds., 29| yds. 

5. Traveling 5 days, at the rate of 18 miles the first 
day, 20 the next, 20^ the next, 22J the next and 25 the 
next, what was the average speed per day? 

Answers : 70, 3% 10, 21^, 21 T V 



AVERAGE. 181 

EQUATION OF PAYMENTS. 

179. When average is applied to the settlement of ac- 
counts, the process is called Equation of Payments or 
Equation of Time. 

Merchants and manufacturers sometimes sell their goods 
on credit, the time varying from three to nine months, 
and their customers making numerous purchases before 
settlement. The object of equation of payments is to ob- 
tain an average date of payment for these purchases. 

6. I owe $3 payable in 2 months, $4 payable in 3 
months and 5 payable in 4 months , what will be the av- 
erage time of payment for the whole amount? 

The use of $3 for 2 mos^that on $1 for 3x2= 6 mos. 

The use of $4 for 3'mos=that on $1 for 3x4=12 mos. 

The use of $5 for 4 mos=that on $1 for 4x5=20 mos. 

$12 38 mos. 

Hence, the use of $12 for those several months is equal 

to that of $1 for 38 months; for 12, it will be -^ as much, 

or 3^ months or 3 months 5 days. 

7. A merchant sells a bill of goods amounting to $4000, 
to be paid as follows: $400 in 30 days, $600 in 60 days, 
$1000 in 90 days and the balance in 4 months, or 120 
days; what would be a mean or average time of payment 
for the whole? 

A credit of $400 for 30 ds. is the same as a credit on $1 for 12000 ds. 
600 " 60 " " " " " 1 " 36000 " 

" 1000 " 90 " " " " " 1 " 90000 " 

" 2000 " 120 " " " " " 1 " 240000 " 

4000 378000 

On $1 there is a credit for 378000 days. 
On $4000, there is a credit for 378000 days divided by 
~94| days. 



182 NELSON'S COMMON-SCHOOL ARITHMETIC. 

That is, the $4000 might be paid in 94J days, or on 
the 95th day, without either party sustaining loss by in- 
terest. 

8. A merchant sells goods to the amount of $1700, $500 
payable in 60 days, $300 payable in 90 days and $900 
payable in 30 days ; what is the average time of payment 
of the whole? 

9. Sold a bill of goods amounting to $700, J- of which 
is payable in 90 days, ^ in 4 months and J- in 6 months; 
required the average time of payment* 

180. To find the average date of purchase. 

10. Purchased goods as follows; what was the average 
date of purchase? 

December 31, a bill of $300; January 3, a bill of $100; 
January 9, a bill of $200, January 18, a bill of $800; 
January 23, a bill of $500. 

300 X EXPLANATION. The first was due at the 

100X 3 300 time of purchase; the second, three days 
200 X 9 1800 after; the third, nine days after, etc. 
800 X 18=14400 REMARK. If the amounts above were equal, 
500X23=11500 an d the intervals also equal, the average date 
of purchase would be on Jan. 9, because it is 



1900 o 

' __ midway between the first and last dates. 

*W 

Or 15 days after December 31, the date of the first pur- 
chase, which brings the time up to January 15. 

If these debts had been contracted on a credit of three 
months, a note dated January 15 would be given to settle 
the bill. 

The time tables (pp. 168, 169) are admirably adapted for 
averaging, prepared as they are for the two common pe- 
riods of settlement, January and July. For account sales, 
the book-keeper should prepare similar ones for each month. 

*For answers, see end of chapter. 






AVERAGE. 183 

11. The following goods were sold on a credit of 90 
days : 

Required the average date of purchase, or date of note. 

Jan. 1, Invoice of Coffee 81000.00 

Jan. 6, " " Sugar 3500.00 

Mar, 9, " " Sunds 9734.00 

Mar. 13, " " " 976.50 

Apr. 3, " " " 1037.00 

16247.50 

Required the date of maturity of a 3 months' note, 
grace included. 

12. Sept. 3, Invoice of Calicoes $3150.00 

" 19, "Muslins 1174.00 

" 20, " " Silks 3500.00 

Oct. 19, " " Sundries 1743.00 

89567.00 

Find the equated time of payment for the following, or 
date of a sixty-days' note : 

13. Apr. 3, 167.25* 14. May 7, $674.40 

9, 374.00 Jun. 7, 168.37 

" 19, 176.00 . " 10, 370.20 

" 20, 371.00 15, 167.00 

" 25, 197.87 " 19, 679.60 

" 30, 300.00 July 23, 679.45 

May 9, 150.57 Aug. 18, 993.18 

" 23, 720.18 " 19, 875.57 

181. Wlien goods are purchased at different dates and 
on different lengths of credit. 

15. Purchased the following bills of merchandise; re- 

* When the cents are under 50, reject them; otherwise, add a dol- 
lar to the dollars. 



184 NELSON'S COMMON-SCHOOL ARITHMETIC. 

quired the average date of maturity, or the equated time 
of payment for all : 

Apr. 3, a bill of $250 on 3 months' credit. 

Apr. 9, " " 157 " 6 " " 

May 7, " " 250 " 4 " " 

Jun. 9, " 320 " 2 " " 

If we substitute the date of maturity of each of these 
bills for the date of purchase, and arrange them in the 
order of time, we shall have a problem in all respects sim- 
ilar to those under last Art. 

The first bill falls due July 3;* the second, October 9; 
the third, September 7; and the fourth, August 9. Ar- 
ranged in the order of time, they appear thus : 

July 3, $250 X 2 500 By the July table we find the 

Aug. 9, 320 X 3912480 difference of time to be 2, 39, 68 

Sep. 7, $250 X 6817000 and 100 from the first, and the av- 

Oct. 9, $157 X 10d 15700 era e time > 46 an(i over a half, or 

47 **?*' 47 da * s after 



977 )45680(46 

3908 August 17. 

6600 
5862 

"738 

When time tables are used, especially such as are 
adopted for periodical settlements, as those in banking, 
the purchases need not be arranged in the order of date, 
as above. 

16. Find the average date of payment for the following: 

Feb. 3, Mdse, 3 mos, $678.59 July 27, Mdse, 30 ds, $1500.00 

Mar. 9, 2 " 243.75 Aug. 9, " 90 " 175.50 

Apr. 13, " 4 1000.00 Oct. 3, 2 mos, 1673.13 

Jun. 17, 30 ds, 976.54 Nov. 18, " 3 " 987.65 

19, 3 mos, 786.15 Dec. 13, 3 685.18 

*Days of grace are not allowed on invoices. 



AVERAGE. 



185 



182. 'When cash goods are sold with others on credit. 

Groods are sometimes classed as cash or time. Those 
designated cash do not always realize present payment, 
any time within a month being considered cash; and even 
at the end of a month cash has not been exacted, the un- 
derstandino- being that a debt contracted on cash terms 

o o 

draws interest from date. When computing average, cash 
bills are considered due on the day of purchase. 



17 



Jan. 1, 
Feb. 6, 
Mar. 8, 
Apr. 17, 



900 X 59=r 53100 
800X125=100000 
700 X 66= 46200 
600X106= 63600 



3000 



JOHN MICKLEBOROUGH. 

To Mdse on 2 months $900.flD 

" " "3 " . . 800.00 

" " " cash 700.00 

" " " 600.00 

EXPLANATION. 2 months from Janu- 
ary 1 is March 1, or 59 days; 3 months 
from February 6 is May 6, or 125 days 
from January 1 ; March 8 is 66 days from 
January 1, and April 17 is 106 days from 
January 1. Hence, March 30 is the aver- 
age date of maturity or payment. 

88 days after Jan. 1 or Mar. 30. 



)262900 



Find the average date of maturity of the following: 
D. E. SOMERSET. A. E. NELSON. 



18 

Jan. 1, On 3 mos, $600.00 

Feb. 3, For cash, 670.00 

Mar. 3, 6n 6 mos, 950.00 

May 3, For cash, 550.00 



July 1, On 3 mos, 
13, On 2 mos, 
19, For cash, 
23, On 5 mos, 



19 

$675.00 
619.54 
147.67 

678.44 



20 Sept. 3, At 30 days, $937.15 

9, At 90 days, 897.78 

17, Cash, 619.18 

Oct. 3, At 60 days, 777.00 

16 



186 NELSON'S COMMON-SCHOOL ARITHMETIC. 

Required the amount due on each of the following on 
July 1 : 

21. Jan. 9, $678.44 at 60 days; 20th, $419.88 at cash 
price; 29th, $789.14 at 3 mos. 

22. April 9, $1678 on 3 mos; June 18, $1000 at cash 
price; 21st, $879.55 on 60 days; 23d, $371.19 cash; 29th, 
$785.25 cash. 

183. Average Applied to statements when there are credits. 

Dr. C. A. WALWORTH. Or. 

1867. 1867. 

Jan. 1, To Mdse, $300.00 Feb. 9, By Mdse, $200.00 

Mar. 3, " 500.00 23, u " 100.00 

300X 200X39=7800 

500=61=30500 Dr. products. 100x53=5300 
gjJo 13100 Or. products. ^ ^^ 

300 500)17400 
Bal. due 500 34|- or 35 days from Jan. 1 or Feb. 5. 

EXPLANATION. Assuming both purchases and sales to be due on 
January 1, W. would be entitled to a discount on his purchases 
equal to that on $1 for 30500 days, and I would be entitled to a 
discount on my purchases equal to that on $1 for 13100 days, mak- 
ing a difference of 17400 days in W.'s favor on the balance, $500. 
The discount on $1 for 17400 days is equal to the discount on $500 
for 17400 days^-500r=34| days from January 1. Whether to bo 
reckoned backward or forward is easily determined. To be in his 
favor, the time for payment must be reckoned forward; otherwise, 
it would be reckoned backward. 

Required the equated time of the following: 

WILLIAM SILLETS. 

24 

Feb. 3, To Mdse, $250.00 Mar. 2, By Cash, $300.00 
Mar. 9, " " 300.00 Apr. 3, " " 200.00 

Apr. 18, 500.00 



AVERAGE. 187 

Dr. J. C. HINTZ. Or. 

1869. 1869. 

July 3, To Mdse, $1000.00 Aug. 1, Bj Cash, $500.00 

7, " " 500.00 13, " 500.00 

Au. 18, " " 250.00 

Assuming the day of settlement to be July 1, we have 

1000 X 2= 2000 Days. 500x31=15500 Days. 

500X 6= 3000 " 500x43=21500 " 
J60 X 48=12000 i^T 3WM Or. products. 
1750 17000 " 17000 Dr. products. 

1000 20000 Difference of do. 

$750 Balance due. 

750)20000(26 ds. EXPLANATION. The sum of the credit pro- 

150 ducts being greater than that of the debit 

~ products, shows that the discount is in my 

favor. Hence, in order to settle on the assumed 
4'~){) 
J date, his payments would be at a discount of 

50 26 days for the balance; but as it would be 

impossible to settle on a past date, we will have to charge him 
interest from 20 days prior to July 1 (June 5) to the real day of 
settlement, whatever that may be, or else take his note, dated June 
6, bearing interest from date. Say the date of the settlement is 
September 1. Interest on $750 from June 5 to September 1 is $11, 
which, added to $750 $761, balance due. 

This balance is 9 cents less than what would be obtained 
by computing interest on both sides of the account to Sep- 
tember 1, and is caused by ignoring the T ^ of a day 26 
days being taken instead of 26^. 

NOTE. Book-keepers sometimes omit the tens of dollars when 
averaging. In the above example the hundreds might have been 
omitted without serious error. 



188 NELSON'S COMMON-SCHOOL ARITHMETIC. 

Dr. WILLIAM P. WALLACE. Or. 

1869. 1869. 

Feb. 22, To Mdse. $500.00 Apr. 3, By Cash, $620.00 

Mar. 28, " " 700.00 July 6, " " 520.00 

Apr. 30, " " 900.00 Sep. 10, " " 900.00 

Juii. 8, " " 600.00 Dec. 1, 650.00 

Assuming January 1 as the day of settlement, we have 
the following formula: 

500X 52= 26000 620X 92= 57040 

700X 86= 60200 520x186= 96720 

900x119=107100 900x252=226800 

600X158= 94800 650x334=217100 



2700 288100 2690 597660 

2690 288100 

10 10)309560 

30956 days, 

Or 84 years, 9 months and 5 days (allowing for leap-years} 

to be counted backward, because the discount is in my 
* j 

favor. 

184. Hence, when the balance of the account and tho 
balance of the discount are both in my favor, I coun 
backward; when the former only, I count forward, ano 
vice versa. 

29. Goods bought on January 9, at 60 days, $1376; on 
13th, $780; on May 3, $3400. Payments made May 3 
$1200; June 8, $3500; equated time required. 

30. Balance of last account brought down, including in: 
terest to July 1 ; goods bought August 9, at 3 months 
$2300; September 3, $1500; December 9, $500; and pay 
ments, October 3, $3000; November 9, $2500; averag. 
date of payment required. 

31 to 33. Find the equated time of the following : 



AVERAGE. 



189 



1869. 

July 3, To Mdse, 6 mos, 
15, " " 3 " 
u 3 tt 

: < Cash, 



H. H. SCHULTZ. 

1869. 



An. 21, 

Sep. 18, 

Oct. 15, 

21, 

27, 

31, 

Nov. 28, 
30, 



Cash, 

Mdse, 3 mos, 
u 4 u 



560.87 
149.50 
2000.00 
396.40 
175.20 
425.16 
100.00 
506.18 
197.45 
321.16 



July 1, By Balance, 127.15 

30, " Accept. 60 ds, 300.00 

Aug. 29, Cash, 460.00 

Oct. 20, " Note, 3 mos, 1000.00 



31, 
31, 

Nov. 30, 
30, 



Cash 

Mdse Ret., 
Cash, 
Balance, 



100.00 

250.00 

450.00 

2144.77 



4831.92 



Dec. 1, 

1867. 


To 


Bal., 


2144.77 

THEODORE LILIENTHAL. 

1867. 


Jan. 


t, 


To 


Balance, 


650.00 


Jan. 


8, 


By 


Mdse, 3 mos 


160.00 


Feb. 


3, 


u 


Cash, 




245.00 




15, 


(t 


it 


6 " 


710.87 




15, 


it 


Note, 


60 ds 


416.87 


Feb. 


H 


u 


u 


2 


910.14 


Mar.18, 


it 


Accept, 30 ds, 1000.00 


Apr. 


16, 


It 


11 


Cash, 


1000.00 


Jun. 


4, 


it 


a 


60 " 


750.14 


June 


8, 


11 


It 


4 mos, 


900.00 




16, 


it 


Note, 


3 mos, 


987.64 




15, 


tl 


tl 


4 


2500.00 




30, 


tt 


Cash, 




500.00 




17, 


tt 


It 


6 " 


1215.00 




30, 


tt 


Mdse, 


abate. 


200.00 




30, 


tl 


Sunds 


i 


700.00 












JULIUS 


WISE. 


1869. 


1869. 


July 


3, 


To 


Balance, 


1500.00 


Aug. 


3, 


By 


Cash, 




1000.00 




18, 


tt 


Mdse 


, 4 mos, 


750.40 


Sept. 


7, 


U 


Acctc 


. 60 ds, 


500.00 


Aug. 


29, 


it 


it 


4 


128.80 


Nov. 


5, 


11 


it 


60 " 


750.00 


Sep. 


30, 


tt 


tt 


4 " 


916.84 


Dec. 


14, 


tt 


Cash, 




2000.00 


Oct. 


10, 


u 


it 


3 


500.00 















30, " Cash, 675.14 

Nov. 18, " Sunds, 564.18 

These exercises may be omitted until the learner has 
studied Exchange. 

188. When payments are made before a note or bill is 
due, to find how long after maturity it should run to bal- 
ance the interest on the advanced payments. 



190 NELSON'S COMMON-SCHOOL ARITHMETIC. 

34. A merchant holds a note of $500 at 6 months. 
Three months before it is due, he receives $100, and one 
month before it is due, he receives $300 ; how long should 
he allow the balance to run to equal the interest on the 
advance? 

The int. on 100 for 3 mos=int. on $1 for 300 mos. 
The int. on 300 for 1 mo r^int. on 1 for 300 mos. 

600 mos. 

Hence, the interest on the advance payments is equal to 
the interest of $1 for GOO mouths ; that is, a balance of 
$1 should have run 600 months, but the balance due on 
the note is 100; therefore, it should run |-{^ months 6 
mouths. 

PROOF. The int. of the $100 for 6 mos=$ 3. 

The int. of $100 (the first pay't) for the 3 mos=$1.50 
The int. of 300 (the sec. pay't) for the 1 mo = 1.50 

Total interest on advance=$3.00 

35. A note of $600 was given January 3, 1868, payable 
in 6 months. 4 months before it was due, $100 was p;iid 
on it, and 3 mouths before it was due, $200 was paid; 
how long in equity should the balance run? 

36. A merchant owes $700, due 8 months from the time 
he contracted the bill ; 5 months afterward, he pays $200, 
and two months after that, $300; how long should the. 
balance remain unpaid? 

37. If I borrow $600 from A at one time, and $500 at 
another, each for 4 months, how long should I lend him 
$1000 to return the favor? 

38. I owe $400, payable in 10 months; at the end of 4 
months I pay $100; 3 months after that, $50; how long 
after the expiration of 10 months may the balance remain 
unpaid ? 



AVERAGE. 191 

39. A owes $1000 due in 6 months; 5 months before it 
is due he pays $200, and 3 months before it is due he 
pays 300 ; how long after the expiration of the 6 months 
may the balance remain unpaid? 

187. Average applied to account sales. 

An account sales is a detailed statement of goods re- 
ceived by a commission merchant, and sold on account of 
another. The person who sends the goods is called the 
shipper or consignor; the person to whom they are sent, 
the consignee, and the goods, the consignment. 

The duty of an agent or commission merchant is to pro- 
cure the best intelligence of the state of trade at the place 
where he does business, including the quality and quan- 
tity of goods in the market, their present prices, and the 
probability of their rising or falling; to pay exact obedi- 
ence to the orders of his employers; to consult their ad- 
vantage in matters left to his discretion; to execute their 
business with all the dispatch circumstances will admit; 
to be early in his intelligence, distinct and correct in his 
accounts, and punctual in his correspondence. 

188. An account sales should state from whom the 
goods were received, or on whose account and risk they 
were sold, the dates and terms of sales, the cash paid for 
freight, drayage, etc., and the various charges, such as in- 

i suraiice, commission, cooperage, etc. The total amount 
received will appear on the right of the account, and the 
charges, etc., on the left, or they may be arranged as in 
the example. 

The difference between the two sides is called the pro- 
ceeds. 

Advances made on goods are charged to the shipper's 
account, not in the account sales. Where goods are sold 
promptly for cash, or on short time, the account sales is 
not averaged. 



192 NELSON'S COMMON-SCHOOL ARITHMETIC. 

COMMISSION HOUSE OF STRAIGHT, DEMING & CO. 

Shipment 18. No. 7828. 

Sales for account of Messrs. Gaff & Baldwin. 

By sundries, 
June 4, T. B. Colgan & Co. @ 60 days, 8 hhds. Sugar. 

1095 1020 

1100 1120 

1080 1240 8965 

1200 1110 896 8069 @ ?& $600.13 

June 6, G. Newton & Co., @ 60 days, 10 hhds. Sugar. 

1080 1040 

1090 1340 

1120 1020 

1240 1100 11440 

1200 1210 1144 10296 @ 6| 8707.85 

June 10, B. Vilgers & Co., @ 60 days, 20 hbds Sugar. 

1060 1240 

1210 1110 

1180 1005 

1055 1285 

1240 1100 

1185 1210 

1300 1325 

1010 1140 

1120 1205 23185 

1205 1000 2318 20867 6^ 1343.31 

$2651.29 

CHARGES. 

June 1, P'd cash st'r Landis for freight, $87.18 

10, Dray. 9 50 , ins. 4 63 , and stor. 9 50 , 23.63 

" Commission and guarantee, 132.56 243.37 

Net proceeds due by equation, Aug. 13, $2407.92 

E. 0. E. 
CINCINNATI, June 14, 1858. 

STRAIGHT, DEMING & Co., 
Per F. JELKE. 



AVERAGE. 193 

Aug. 3, 600X63^37800 June 1, 87 

" 5, 708X65= 46020 " 10, 156X9=1404 

9, 1343X69= 92667 ^ ^ 

2651 )176487 

243 1404 

2408 175083(72.6 or 73 days from June 1, 

16856 which gives August 13. 
6523 

4816 

1707 

2. Find the equated time of payment of the following: 
Account sales of merchandise sold on account and risk 

of Morris J. Parry, New York: 

Mar. 1, To Cash for freight, $50.00 Mar. 15, By Cash, $450.00 

1, " Drayuge, 10.00 Apr. 3, " S. Miner, 75.00 

10, " Insurance, 8.50 May 1, " Cash, 31rf.75 

June 20, " Storage* Advertising, 10.00 7, "J.Clark. 92.25 

" Com. on $1224 @ 2,^, 30.60 Jun. 19, " Cash, 288.00 

M. J. Parry's net pro., 1114.90 

$1224.00 $1224.00 

3. Sales of 100 bbls. of molasses for acct. of C. H. 
Crane. 

July 3, F. M. Peale, on acct., 20 bbls., 860 gals... @ 1.25 1075.00 

July 9, Saml. A. Butts, Jr., cash, 15 bbls., 635 gals. @, 1.20 762.00 

July 18, Geo. T. Ladd, cash, 12 bbls., 495 gals @ 1.20 594.00 

Sep. 6, F. M. Peale, on acct., 12 bbls., 495 gals... @ 1.20 594.00 

Oct. 3, J. J. Marvin, on acct., 31 bbls., 1285 gals. @ 1.00 1265.00 

Dec. 18, J. J. Marvin, on acct., 10 bbls., 400 gals.. @ .90 360.00 

$4650.00 
CHARGES. 

July 1, Cash paid freight 58.00 

Drayage 18.00 

Dec. 30, Cooperage 3.20 

Com. and guar. 4 percent 186.00 

C. H. Crane's net proceeds 4384.80 

$4650.00 
17 



194 NELSON'S COMMON-SCHOOL ARITHMETIC. 

SALES FOR ACCT. OF R. H. LANGDALE. 
Sept. 3, G. F. Sands, 90 days, 

18 bbls. Clo. S. 3120 Ibs net=52 bus. @ $6.50 338.00 
Oct. 9, C. F. Howe, on 60 ds. note, 

5 bbls. Clo. S. 930 Ibs., 15J bus... @ 10.00 155.00 
Dec. 5, G. A. Voige, 30 ds. note, 

10 bbls, 2220 37 @ 8.00 296.00 789.00 

CHARGES. 

Aug. 15, Cash pd. fr't 12.00 

Drayage on acct 6.00 

Sept. 1, Advertising 8.55 

Dec. 30, Storage 10.00 

Com. and guar., 5 per cent 39.45 

R. H. Langdale's net proceeds 713.00 799.00 

E. E. 

CINCINNATI, Jan. 1, 1867. NELSON, NEPHEW Co. 

Answers: 3 mos. 24 ds.; 3 mos. 10 ds. ; 3 mos. ; Deo. 
31, 1869; 4 mos. 15 ds.; July 15, 1869; Dec. 15, 1867; 
$1919.85; 4 mos. 12 ds. ; Oct. 17; March 25; $4707.34; 
Nov. 2; 49 ds. ; Feb. 21; July 11; April 30; Dec. 23; 
143 ds. ; April 27 ; Dec. 6 ; Oct. 8 ; May 17 ; Dec. 21 ; 
Aug. 24; June 2; Aug. 16, 1867. 

ACCOUNTS CURRENT. 

An Account Current is a statement of the entire trans- 
actions between two parties, generally for three or six 
months. It exhibits the whole sums given and received, 
the interest due on each at the date of the account, and 
to whom the balance of interest an9 principal is due. 

Without a knowledge of book-keeping, an account cur- 
rent or account sales can not be fully understood. 

The name of the party against whom the account cur- 
rent is made out is always written first and on the left, 
while the maker's name appears on the right. When any 
sums fall due after the date of the account, the interest ia 
entered on the opposite side, making it discount. 






AVERAGE. 



195 



AN ACCOUNT CURRENT WITH INTEREST COMPUTED. 

MESSRS. GRAFF & BALDWIN, 
In Account Current and Interest Account with 
Dr. STRAIGHT, DEMING & Co. 


DATE. 


DKSORIPTION. 


TIME. 


IN'T. 


AMOUNT. 


1867. 
Jan. 
tt 

Feb. 

Mar. 
Apr. 


4 
10 

8 

20 
2 
1.0 
18 


To Mdse, as per bill rendered 


Days 
161 
155 

63 
114 
104 
96 
57 
55 
44 
49 

8 

37 
20 
9 
6 

145 
137 

88 

77 

65 
37 
16 


11 

20 

19 
2 

4 
4 
1 
8 






4 


c. 
40 
93 

25 

00 
17 
08 
12 
02. 
10 i 
99 

67i 

03 

92 
74 
15 

17! 


425 
810 

500 
1000 
125 
5 
434 
504 
150 
1101 

500 

5 
275 
495 
4152 
5 
30 


c. 

00 
00 

00 
00 
00 
00 
00 
00 
00 
00 

00 

00 
00 
00 
00 
00 
03 


tt It U (( 


" Your draft on us at 60 days sight, 
in favor of II King & Co 


" Your sight draft on us 


(t Mdse, as per bill rendered . ... 


" Exchange on $500 at 1 per cent.... 
" Mdse, as per bill rendered 


tt 
May 

June 

u 

Jan. 
Mar. 
tt 

Apr. 
May 

u " 

June 


20 

25 
26 
4 

8 

25 
5 

8 
14 

20 

28 
18 

29 
10 

8 
29 
14 


" Our draft on G. Wright, your favor. 


" N. Davis & Co.'s note at 30 days, 


tf Discount on uncurrent money at 
1/1* P er cent 


l< Mdse, as per bill rendered 


" Net proceeds, sales 100 bbls. flour. 
" Mdse, as per bill rendered 


" Cash paid for vour telegram 


" Interest in our favor 


CR. 

By draft on !New York, % P- ? prem. 
" Cash 


84 

24 
2 

11 

5 

6 
1 
2 

30 

84 


10515 


03 


29 
28 

74 
17 

76 
23 
67 

03 
17 


1005 
100 

800 
403 

624 
200 
1000 
2407 

3976 
10515 


00 
00 

00 
00 

00 
00 
00 
00 

03 


" Your sight draft on Moore & 
Adams, our favor 
" Draft on New York, % P er cent. 


" Net proceeds, sales 50 bbls. ino- 


Cash 




" Net proceeds, sales 38 hhds. sugar. 




E & E 


CINCINNATI, June 14, 1867. 
STRAIGHT, DEMING & Co., 
per HILL. 


03 



196 NELSON'S COMMON-SCHOOL ARITHMETIC. 



XXII. EXCHANGE. 

189. IF A of Liverpool is indebted to B of New York 
in the sum of $1000 dollars, and B is indebted to C of 
Liverpool to the same amount, it is evident that the two 
debts might be paid without either party sending real 
money to the other. A settlement could be effected by B 
sending the following order to C : 

$1000. NEW YORK, Jan. 3, 1868. 

At sight, pay to the order of C One thousand dol- 
lars, value received, and charge to my account. 

To A, B 

Liverpool, England. 

This method of settling accounts between persons in 
distant places is called Exchange. If between persons in 
the same country, it is called Home, Domestic or Inland 
Exchange; otherwise, it is called Foreign Exchange. 

190. A Bill of Exchange is an order written like the 
one above ; it is also called a check or draft. The term 
draft is usually applied to inland bills. There are Sight 
Bills and Time Bills. 

191. A Sight Bill is one payable at sight or on de- 
mand, as the one above. 

A Time Bill is one that requires payment so many 
days after sight, or after date. 

192. A Set of Exchange consists of three copies of the 
same bill drawn to insure safety of transmission, one of 
which being paid, the others are void. 

The person whose signature is attached to a bill or 
draft is called the drawer or maker; the person addressed, 
the drawee; the party to whom payment is to be made, 






EXCHANGE. 197 



the payee, and the one who has possession of it ? the owner 
or holder. 

REMARKS ON BILLS AND DRAFTS. 

1. Bills of exchange, unless payable at sight or on de- 
mand, require to be " accepted," and should be presented 
promptly to the drawee for this purpose. The drawer 
then becomes the acceptor, and the bill is said to be ac- 
cepted or honored, and is called an acceptance. 

2. Accepting a bill consists in the drawee writing his 
name across the face, by which act he becomes responsible 
for its payment. The following is the form: 



&C CO. CO 



Business men prefer a draft to a promissory note, be- 
cause there are three parties to it, while the note, ordi- 
narily, has only two. It often happens, however, that 
notes and drafts are made payable to the drawers, which 
leaves only one party to the former, and two to the lat- 
ter, though technically they are considered the same as if 
drawn in favor of others. 

When bankers receive unaccepted bills, they send them 
out for acceptance or notify the drawees. 



198 NELSON'S COMMON-SCHOOL ARITHMETIC 

Bills of exchange, like promissory notes, may be made 
negotiable by the insertion of the words, " to the order 
of," or "bearer," and are subject to protest for non-pay- 
ment, and the indorsement may be special or in blank. 

3. Bills of exchange can be had at the banking houses 
or offices of exchange brokers, and may be drawn in favor 
of the persons buying them, indorsed and mailed to their 
creditors; or they may be drawn in favor of some other 
person, indorsed by him, and afterward indorsed by the 
buyer, and mailed to his creditors, as before. 

4. When a bill or draft costs neither more nor less than 
the amount of its face, it is said to be at par; if less than 
that amount, it is at a discount; if more, it is said to com- 
mand a premium, and the rate of discount or premium is 
called the rate of exchange. 

5. The phrase, "apply to my account," or "your ac- 
count," "as advised," etc., are not essential to a bill, but 
rather indicate the relations of the parties as debtors or 
creditors. When the drawer is indebted to the drawee, 
he would say, " apply to my account," but when the drawee 
is indebted to the drawer, the phrase would be, " apply to 
your account," or "put it to your account." Should the 
bill be drawn on account of a third person, he would say, 
"put it to the account of A." 

6. When the words "as per advice," or "as advised," 
are used, it is presumed that a letter of instructions has 
preceded the draft. In such case, the drawee honors at 
his risk in the absence of such advice. 

7. Bills or drafts for acceptance must be presented 
within a reasonable time. If the drawee destroy a bill for 
acceptance, or refuse to return it in twenty-four hours, he 

be deemed to have accepted it. 



EXCHANGE. 199 

8. Sight bills for collection should not be mailed to the 
drawee, as their possession is presumptive evidence of 
payment. 

9. The phrase "value received" is properly omitted 
when the bill is drawn against funds of the drawer in the 
hands of the drawee, as is usually the case with banking 
houses when selling exchange. 

10. The place of payment, separate from where it is 
drawn, is not usually inserted in a draft, unless an under- 
standing to the contrary exists between the parties. 

11. Drafts are often drawn by merchants upon each 
other to raise money or settle accounts. A merchant 
shipping a large quantity of goods to another to sell on 
commission, usually draws a draft for a part of the cost 
on the party and sells it at bank, or passes it to another 
merchant in the course of business. This kind of paper 
is called a mercantile draft, to distinguish it from one is- 
sued by a bank, which is called exchange, or a bank check 
or draft, and is not so available for transmission as the 
bank draft or exchange. It is a part of the business of a 
banking house or exchange office to buy this mercantile 
paper, send it home for collection, and in the mean time 
sell exchange on the banks to which they transmit it, for 
such sums as may be demanded. 

DRAFT OF A MERCHANT UPON ANOTHER TO WHOM HE 
HAS SHIPPED GOODS. 

NEW YORK, May 17, 1867. , 

At ten days' sight, pay to our order One thousand dol- 
lars, value received, and charge to our account. 

To HENRY L. WEHMER. A. J. RICKOFF & Co. 

Cincinnati, Ohio. 

To obtain money on this, Mr. Rickoff would indorse it 
to a banking house, which would pay him the current 



200 NELSON'S COMMON-SCHOOL ARITHMETIC. 

rate for mercantile paper, and discount for the time to ex- 
pire before collection say 4 days to reach its destination, 
and 13 days for maturity. 

BILLS IN DUPLICATE AND SETS OF EXCHANGE. 

To prevent delay and guard against loss, bills of ex- 
change are often drawn in duplicate or in sets.* 

A duplicate bill is, as the name indicates, a correct copy 
of the original, with the addition of the word " Dupli- 
cate" written or printed across the face.f 

A set of exchange properly belongs to Foreign Ex- 
change, under which subject the learner will find the form 
and description. 

HOME EXERCISES IN DRAWING DRAFTS. 

1. Draw on your teacher, (locating him in San Fran- 
cisco, California,) at 10 days' sight, for $2136.50, and make 
the draft payable to yourself. 

2. At 60 days after date, draw on 0. I. Mitchell for 
$2000, favor of yourself, and prepare draft for negotiation. 

Draw bills and notes from the following data: 

3. Draw on M. Garaghan, St. Louis, for $3600, at sight, 
and prepare the draft for collection by the Central Na- 
tional Bank, Cincinnati. 

* Should a bill be lost in transmission, the amount can be recov- 
ered of the bank from which it was bought, unless it can be proved 
that payment was made by the drawee. 

t On bank drafts will often be found writing on both back and 
face, which can not be represented in type, such as the names of 
bank officers, through whose hands, they pass before issue, the 
amount written on the back or face a second time to prevent alter- 
ation. They are also drawn payable in "gold" or "currency," as 
occasion requires. 



EXCHANGE. 



201 



Drawer, 

Drawee, 

Payee, 

Date, 

Where, 

Time, 

Amount, 



4. 5. 

Yourself. Drawer, Yourself. 

Your teacher, at N. 0. Drawee, W. A.Fillmore,N.Y. 



G. A. Carnahan. Date, 

The present. Time, 

Cincinnati, Ohio. Payee, 



April 1, 1867. 
10 days' sight. 
Yourself. 



At sight. 
$367.25. 



Amount, $5907.84. 



6. Face, $3167.85. 

Date, August 9, 1866. 

Payee, B. 0. M. De Beck. 

Indorsee, Luther W. Strafer. 

Drawer, Noble K. Royse, Cincinnati, Ohio. 

Drawee, Herman H. Raschig, New Orleans. 

Time, 10 days' sight. 

7. ACCEPTANCE OF F. M. PEALE AT NEW YORK. 

Drawer, Wm. H. Morgan, San Francisco. 

1st Indorser, J. M. Allen. 

Time, 60 days after date. 

Face, $967.18. 

Date, 24th May, 1866. 

Indorsee, Geo. F. Sands. 

8. Drawer, John J. Marvin, Cincinnati, Ohio. 

Drawee, John S. Highlands, Columbus, Ohio. 

Payee, E. H. Prichard, Boston, Mass. 

1st Indorser, B. B. Stewart, Face, $2127. 
Time, 30 days after sight. 



Date, 



January 1, 1867. 



Accepted two days afterward. 

9. Indorsee, G. W. Harper. 

Drawee, C. R. Stuntz, Washington, D. C. 

Drawer, G. W. Smith, New York. 

Indorser, G. A. Schmitt. 

Time, 10 days' sight, Face, $5000. 

Date, January 1, 1867. 
Accepted four days after date. 



202 NELSON'S COMMON-SCHOOL ARITHMETIC. 
AN INDIANA NOTE. 

$500. INDIANAPOLIS, May 25, 1867. 

Six months after date, I, the subscriber, of Indianapolis, 
County of Marion, State of Indiana, promise to pay to the 
order of Geo. W. Runyan, Five hundred dollars, without 
any relief whatever from valuation or appraisement laws. 
Value received. Payable at First National Bank. 

JAMES W. WILSON. 

No. 59. Due November 28. 1867 



XXIII. FRACTION'S. 

193. A FRACTION is a part of one or more things con- 
sidered as a whole, and is therefore the result of division. 

Whole numbers are sometimes expressed in fractional 
form. (See Improper Fractions.) 

194. The subject of fractions is the method of treating 
fractional numbers, or showing how they may be added, 
subtracted, multiplied and divided. 

195. Fractions are divided into common and decimal, 
according to the way in which they are written. A com- 
mon fraction requires two numbers to express it, as J, 
while a decimal requires only one, with a period at the 
left, as .5. 

The following fractions would be read as shown oppo- 
site: 

jiy One Twelfth. 

fr Three Seventeenths. 

Nine One hundred and forty -fifths. 

Fifty seven Three thousand three hundred and 
ninety -sixths, or Thirty-three hundred, etc. 



FRACTIONS. 203 

198. The two numbers forming a fraction are called 
terms; the upper term the numerator, and the lower term 
the denominator. The line between the terms is the sign 
of division, and indicates that the upper term is divided 
by the lower. (Art. 34.) -f% represents the twelfth part 
of 3, or 3 parts of something divided into 12 parts. 

A fraction also expresses the ratio between the two 
terms. ^=3 : 12. 

PRINCIPLES OF FRACTIONS. 

197. If both terms of a fraction be multiplied by the 
same number, the value of the fraction will remain un- 
altered. 

Let the 1 and 4 of the fraction J be multiplied by 2, 
and we have -|, a fraction of the same value as J. 

198. If both terms of a fraction be divided by the 
same number, the value of the fraction will remain un- 
altered. 

Let 3 and 6 of the fraction - be divided by 3, and we 
have ^, a fraction of the same value as -|. 

199. If the numerator only be multiplied, the value of 
the fraction will be increased and the whole fraction mul- 
tiplied. 

Let 2 of the fraction f be multiplied by 4, and we have 
|; that is, eight thirds instead of two thirds. 

200. If the denominator only be multiplied, the value 
of the fraction will be decreased, and the whole fraction 
divided. 

Let the 3 of the fraction be multiplied by 2, and we 
have | ; that is, two sixths instead of two thirds. 

201. If the numerator only be divided, the value of the 
fraction will be decreased, and the whole fraction divided. 

Let 4 of the fraction |- be divided by 2, and we have 
|-; that is, two eighths instead of four eighths. 



204 NELSON'S COMMON-SCHOOL ARITHMETIC. 

202. If the denominator only be divided, the value of 
the fraction will be increased, and the whole fraction mul- 
tiplied. 

Let 8 of the fraction -| be divided by 2, and we have 
J ; that is, two fourths instead of two eighths. 

Common fractions are divided into simple, proper, im- 
proper, contfpvund, complex fractions and mixed numbers. 

A simple fraction is a single fraction, as f . 

A proper fraction is a single fraction whose numerator 
is less than the denominator, as jr. 

An improper fraction is a single fraction whose numer- 
ator is equal to or greater than the denominator, as f, f , 
which indicates not a part, but the whole or more than 
the whole; hence, the term improper. 

203. A compound fraction is a fraction of a fraction or 
fractions, as % of f or 4 of -J-* of ||. 

204. A complex fraction is one having a fraction in the 
numerator or denominator, or in both, as 



205. A mixed number is composed of a fraction and 
whole number together, as 7f . 

REDUCTION OF FRACTIONS 

206. Fractions are often expressed in terms too large 
for convenient use, or to estimate their value at sight. 
The fraction -^ possesses the same value as J, and for 
convenience in operating ought to be reduced to that de- 
gree of simplicity. 

The process of changing the form of a fraction in this 
manner is called reducing it. 

207. To reduce a fraction to its lowest terms. 



FRACTIONS. 205 

We divide both terms by any number or numbers which 
will do so without a remainder. (Art. 198.) 

1. 9 and 27 of the fraction ^ divided by 9, give 1 
and 3, or the fraction ^. 

When a single number will not reduce a fraction to its 
lowest terms, other numbers are used and the process con- 
tinued.* 

2. To reduce -- to its lowest terms. 



< 

3 to 13. Reduce the following fractions to their lowest 
terms: 
H> 4tt. TS, iWr. T#&. T&V&. dMr. isWff. If!!. 

9 3 6 9876 
2"8lT8> TSTTOT- 

Answers : |, |, yffj-, ^fff , |, 3 \, |, f f , #& i dhr- 

208. ^ raise a fraction to a higher denomination, we 
multiply both terms by the same number a process the 
reverse of the last. 

209. When the higher denomination is given, the mul- 
tiplier may be obtained by dividing the new denominator 
by the old. 

*THE GREATEST COMMON DIVISOR 

Is the greatest number which will divide any two or more numbers. 
It may be found by the following process, but the operation is so 
long that it is seldom used in practice. 

To find the greatest common divisor of 540 and 612. 
640)612(1 EXPLANATION. The smaller number is di- 

540 vided into the larger, and the remainder (72) 

72)540(7 into the first divisor; then the next remainder 

504 (36) into the last divisor, etc. 

36)72(2 The last divisor is the greatest common di- 

7^ visor, viz., 36. That is, no number higher than 
36 will divide both without a remainder. 



206 NELSON'S COMMON-SCHOOL ARITHMETIC. 

14. To raise ^ to 24ths, we divide 24 by 6, which gives 
4 as a multiplier. 5x420 an d 6x4=24, making |f. 

15 to 21. Raise f to 12ths, $ to IGths, -f to 49ths, | to 
28ths, | to 120ths, -^ to ISOths, ff to 500ths. 

210. To reduce a mixed or whole number to an improper 
fraction. 

22. Reduce 5|- to an improper fraction ; that is, in 5-J- 
how many eighths? 

5? EXPLANATION. In every whole number there are 8 eighths, 
8 and in 5 whole numbers there are 8 times 5, or 40 eighths, 
"TT to which add 7 eighths and the result is 47 eighths. 

-r 

23 to 30. Reduce the following numbers to improper 
fractions: 15f, 7f 6$, 51f, 17, 113ft, U, 21 f 
Answers : -*f<s fy M, ^, *f> Ai f> S ^^ 

211. ^o reduce improper fractions to whole or mixed 
numbers, is an operation the reverse of the last. 

31. Reduce ^ to a mixed number. 
8)47 

~&i 

32 to 41. Reduce the following to whole or mixed num- 
bers, and the remaining fractions to their lowest terms: 

"t 4 . H*. -f A > W. Hf 6 -> - x r. -H 6 - -. ^iW, WA H?- -- 

Answers: 182, 10ft, 2% 50^, 12^ 693^, 17#V. 



XXIV. DECIMALS. 

212. A DECIMAL FRACTION expresses its value in one 
term, and is known from a whole number by its having a 
period, called a decimal point, at the left. .5 is a decimal. 

The value of a decimal is more easily ascertained than 



DECIMALS. 207 

that of a common fraction, while operations in decimals 
are performed with nearly the same ease as those in whole 
numbers. 

213. Figures increase in a tenfold ratio as they are re- 
moved one place to the left, and decrease in the same 
ratio as they are removed one place to the right. 

In the number .5 the figure is one place to the right 
of the unit figure, and therefore possesses only one-tenth 
of the value it would in that place. In other words, it 
represents tenths instead of units. One place further it 
would represent hundredths, as .05, and one place further, 
thousandths, as .005. As common fractions these would 
appear thus : ^, yjfo T f^ ; hence, 

214. To reduce a decimal to a common fraction, we 
erase the decimal point, and write for the denominator as 
many ciphers as there are figures in the numerator, and 
prefix the figure 1. .075 would be written yj-g-g- and 
0047, TD Vw- 

Ciphers on the extreme right of a decimal possess no 
value. .500 expresses the same value as .5, the first be- 
ing -j^nnr, which, reduced to tenths, is 



1 to 21. Find the fractional value of the following and 
read them: .23, .007, .013, .760, .00019, .3401, .67800, 
.0907, .0076, .3467, .1093, .0770, .3657, .2136, .09876, 
.000001, .13607, .06789, .03146, .000016, .016037. 

215. As in Federal money, the removal of the decimaJ 
point one place to the right multiplies a number by 10, and 
its removal one place to the left divides it by 10. 

Removed to the right in 1673.27, we have 16732.7, and 
removed to the left we have 167.327. 

The notation and numeration of decimals being so sim- 
ilar to that of whole numbers, little trouble will be expe- 
rienced in reading the following: 



208 NELSON'S COMMON-SCHOOL ARITHMETIC. 



22. .01 


28. .3013 


34. .45689 


40. .00001 


23. .57 


29. .0031 


35. .18654 


41. .0010906 


24. .709 


30. .2160 


36. .36109 


42. .3016031 


25. .856 


31. .1061 


37. .00009 


43. .0016039 


26. .2913 


32. .4064 


38. .01002 


44. .0000067 



27. .0016 33. .5067 39. .168002 45. .00001001 

Express in figures the following: 

1. One tenth. 
- 2. Three hundredths. 

3. Five thousandths. 

4. Sixty-five ten thousandths. 

5. Three hundred seventy-six thousandths. 

6. Four hundred ten thousandths. 

7. Eighteen hundred and twenty thousandths. 

8. Sixty-one fand eighteen hundredths. 

9. Forty-five hundredths. 

10. Eighty-seven thousand jand sixty hundredths. 

11. Five hundred thousand and seven tenths. 

12. One hundred and one thousand and seven. 

13. Sixty-four thousand and eight tenths. 

14. Nine millions. and seventy-nine thousandths. 

15. Eighty-six hundredths. 

16. Seven thousand and six hundredths. 

17. One hundred and ten and sixty-five hundredths. 

18. Eighteen hundred and sixty-seven^and seventy-five 
hundredths. 

19. Twenty-four hundred and five hundred and one 
thousandths. 

ADDITION OF DECIMALS. 

216. When arranged, tenths under tenths, hundredths 
under hundredths, etc., decimals are added and subtracted 
precisely as whole numbers. The operations in Federal 



DECIMALS. 209 

money, with which the learner is already familiar, properly 
belong to this subject. In those, the decimal points were 
placed directly under each other. The same rule should 
be observed in adding or subtracting decimals generally. 

I. To add 1.07+.001+37.045+10.06+.0007. 

1.07 EXPLANATION. Here the decimal points are arranged 

.001 directly under each other and addition performed as in 

37.045 whole numbers. 

10.06 
.0007 

48.1767 

2. 2.13 + .426 + 21.2 + 7.63 + 640.072=? 

3. 43.27 + 9.042 +712.417 + 41.007 + .962=? 

4. 820.71 + 2.006 + 84.243 +217.072 + 9.341=? 

5. 107.67 + 1.301 + 20.0163+684.6 + 10.06 =? 

6. 719.86 + .2103+ .1610+310.6 +2134. =? 

7. 9.8784+29.8 + 67.19 + 7.916 + 379. =? 

8. 643.72 + .109 +360.06 + .0006+ .216=? 
Answers: 2748.3678, 671.458, 1133.372, 823.6473, 

806.698, 1004.1056, 493.7844, 2748.3678, 2708.3768, 
493.7844, 3164.8313. 

9. .007+31.06 + .1009+100.07 =? 

10. 710.34 + 2.406 +67.709 + .0006=? 

II. 314.60 + .0006+ .0027+ .001 =? 
12. 714.06 + .003 + 8.007 +800 =? 

Answers: 314.6043, 1522.07, 131.2379, 780.4556, 
1522.3074. 

SUBTRACTION OF DECIMALS. 

When the larger number has fewer places of decimals 
than the smaller, the blanks may be filled with ciphers. 
(Art. 214.) 
18 



210 NELSON'S COMMON-SCHOOL ARITHMETIC. 

1. To find the difference between 107.06 and .213. 

OPERATION. 107.060 
.213 

106.847 

2. 617.07 41.7106:=? 6. 341. .213 =? 

3. 10.06 .9092=? 7. .97 .0376=? 

4. 36.84 6.672 =? 8. 4.15 .1999=? 

5. 118.09 7.009 =? 9. 7.96 .9789=? 
Totals, 351.6056, 725.7592, 352.6506. 

MULTIPLICATION OF DECIMALS. 



217. In this rule we multiply as in whole numbers, 
and mark off as many places of decimals in the product 
as there are in the two factors. 

1. To multiply 6.107 by 5.5. 

6.107 There are three places of decimals in this factor, 
5.5 and one place in this; 

30535 
30535 



33.5885 so we point off four in the product, 
PROOF. 6.107X 5=30.535 
6.107X.5= 3.0535 
33.5885 

EXPLANATION. The 6 and 107 thousandths multiplied by 630 
and 535 thousandths, but multiplied by .5 or 5 tenths, it is only one 
tenth as much, or 3.0535 (Art. ^13), which, added to the first product, 
gives 33.5885, as above. 

2. .3507 X 10.09 =? 7. 2300.7 X 48.003 =? 

3. 17.07 X200.6 =? 8. 704.23 X .0007=? 

4. 785.4 X 36.70 =? 9. .786x100 =? 

5. .279 X160.7 =? 10. 4.862X .75 =? 

6. 876.5 X .780=? 11. 200.03 X .002 =? 

Total, 32980.4659. Total, 110523.641621. 



DECIMALS 211 

218. When the product contains fewer figures than there 
are decimals in the factors, the number is made up by pre- 
fixing ciphers. 

12. 100X.0005=? 

100 
.0005 

500, to which prefix one cipher and we have .0500, or 
.05, the answer. 

PROOF. .0005X100=.05. 

13. .107 X-05 ==? 16. .3045X.00061=? 

14. 61.04 X-0007=? 17. .27 X-27 =? 

15. .7103X-004 ==? 18. .4102X-1004 =? 
Totals, .0509192 and .114269825. 

DIVISION OF DECIMALS. 

219. When dividing decimals, the quotient and divisor 
must contain as many places of decimals as the dividend. 

1. 33.5885-=-6.107=r? 

6.107)33.5885(5.5 PROOF. This is the converse of Ex. 1 in 
30 535 multiplication, the multiplier and multipli- 

cand being 6.107 and 5.5 and the product 

_ 



30535 

A further proof is obtained by estimate, if we divide the whole 
number (83) of the dividend by the whole number (6) of the divisor, 
which will give one pla-ce for whole number. 

220. When the dividend does not contain as many 
decimals as the divisor, ciphers may be annexed to make 
up the number. The quotient will then be a whole num- 
ber, as it simply shows the number of times the latter is 
contained in the former.* 

*In practice, decimals are seldom carried to more than four places. 



212 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



2. 3066-r-.1783=? 

.1783)3066.0000(17195.73752 



1783 

12830 
12481 

3490 
1783 



17070 
16047 

10230 
8915 

13150 
12481 

6690 
5349 



In this example four ciphers have been an- 
nexed to the dividend, to correspond with the 
number of decimals in the divisor. From 
this resulted 17195, the quotient. These ci- 
phers being exhausted, five more were an- 
nexed to the remainders to give the five deci- 
mals in the quotient. 

Another method is to annex ciphers at 
will, observing to place a mark in the divi- 
dend to limit the whole numbers in the quo- 
tient, as 3066.0000000. 



13410 
12481 

9290 
8915 



3750 

3566 

~184 

221. When there are not figures enough in the quotient to 
make up the number of decimals in the dividend, ciphers 
should be prefixed to the former. 

3. Divide 10.70067 by 370.4. 

370.4)10. 70067(.0288 Here the quotient produced only three 
7 408 figures (288), which, with the one in the 

. divisor, makes only four decimals, so to 

make U P tn e number a cipher is prefixed. 



32947 
29632 



DECIMALS. 213 

Carry out the following to only four places of decimals : 

4. 314.06 ~ 10.73 =? 8. 6.742.34 ==? 

5. 17600 -f-785.4 =? 9. 496 -*- .278=? 

6. 3170.09 -r- 2.4014? 10. 7.6 -5- .734=:? 

7. 417.456-5- 31.145 ==? 11. 7.23--4.06 =? 
Totals, $1385.1824 and 1799.1878. 

12. 30.640 -493.67 = ? 16. 724.1 -f-38.07 =? 

13. 10.8739-^-117.406=? 17. 82.03 -~- 9.0002 =? 

14. 6.342 ~ 22.973=? 18. 7.624 -5- 2.001 =? 

15. 1467.06 -=-196.04 = ? 19. .5213-7- .24121=? 
Totals, 7.9142 and 34.10573. 

REDUCTION OF DECIMALS. 

222. To reduce a Common fraction to a Decimal. 

1. Reduce J to a decimal. 

2)1.0 By annexing a decimal point and a cipher, the number 
7 is properly reduced to tenths or 10 tenths, in which 2 is 
contained 5 times. This 5, being of the same denomi- 
nation of the dividend, is tenths, or .5. 

2. Reduce J to a decimal. 

3)1.00000 This quotient may be carried out indefinitely, 

oqo oq an d i g called a repeating decimal. To save writing, 

a point is usually placed over the repeater thus .3. 

223. The fractional value of a repeating decimal may 
be restored by using 9 instead of 10 as the denominator, 
as f =f 

3. Reduce ^- to a decimal. 
7)1.000000000000 

142857142857+ This is called a circulating deci- 
mal and is marked thus : J42857=|ff||-=f 

Its fractional value is restored in the same manner as 
that of the .3 in the preceding example. 



214 NELSON'S COMMON-SCHOOL ARITHMETIC. 

Express the following decimally:* 

4. J 8. A I*. H 16. A 20. 

5. i 9. ff 13. ^ 17.. & 21. 

6. j 10. ^ 14. ^ 18. 43 22. 

7. f 11. J 15. H 19. ^ 23. 
Answers: .75, .4, .25, .375, .1923+, .6043, .45, .16, 

.6875, .416, .2187+, .046875, .9, .475, .00224, .46, 183, 
.6, .02, .5375, .2. 

224. To find the value of the decimal part of a com- 
pound number, as 0.75 or &0.33J. 

24. What is the value in shillings, pence and farthings 
of 0.345? 

.345 EXPLANATION. Multiplying .345, that is, 

20 fbVu by ^' tlie number f shillings in a 

' or 6 ' 900 



^ , [A ^ n ^ ^ d muhiplying the 900 shminga 

by 12, the number of pence in a shilling, we 



10.800 Pence have Uffffi or 10.800 pence. Multiplying 

4 the .800 by 4, the number of farthings in a 

3.200 Farthings P enn y> we have TSa8 or 3 - 200 or 3 ' 2 farth ' 
or 6s lOd 34- far. in S 8 - 

The operation might have been abbreviated by dropping 
the ciphers on the right. 

Hence, we have for the result 6 shillings 10 pence, 3 
farthings and T 2 or ^. 

Find the value of 

25. .625 of a gallon. 28. .1374 of a ton. 

26. .1425 of a year. 29. .0037 of a Ib. Troy. 

27. .8323 of a . 

Answers: 21 T %; 2, 1; 274, 12 T ff ; 16, 7f ; 1, 21 T V 
225. To reduce denominate values to decimals. 

*The plus sign will be used when the decimal can be carried out 
further. 



DECIMALS. . 215 

30. Reduce 6 shillings 10 pence 3^ farthings to the dec- 
i nal of a pound sterling. 

5) 1.0 The first step in this operation is 

j\ o 9 q <2 / ^ re duce 7= to a decimal, which 

f T' gives .2. Prefixing 3 farthings, we 

12)10.8 o?* lO^j- pence, divide by 4, the number of farthings 

f>(\\ R a A 9 cJiV77 ina penny, and obtain .8 of a penny. 

^jU ) O.5J Cf U^ g b/illl/. r * . 

^ to which we prefix 10 and divide by 

.345 or i'O.-j^o- 12, the number of pence in a shil- 
ling and obtain. 9. Prefixing 6 shillings, we divide by 20, the num- 
ber of shillings in a pound, and obtain .345 of a pound, the answer. 
The reverse of this process is found in example 24. 

The pupil can prove his calculations by last Art. 

31. Reduce 3 quarters to the decimal of a yard. 

32. Reduce 6 Ibs. 3 oz. to the decimal of a cwt 

33. Reduce 12s. 6fd. to the decimal of a . 

34. Reduce 12 Ibs. to the decimal of a tun. 

35. Reduce 1 ft. 3 in. to the decimal of a yard. 

36. Reduce 16 oz. to the decimal of a ton. 

37. At 56 cents a pound, what will 127 Ibs. 6 oz, of tea 
come to? 

16)60 

.375 decimal part of a pound. 

127.375 
56 

764 250 
6368 75 

7133 000 cents, or $7133. 

REMARK. This is not strictly a practical question, nor the short- 
est method of computing the above, the object being merely to show 
the application of decimals. 

38. At $5 for a pound sterling, what will be the value 
of 16 8s. 10d.? 



216 NELSON'S COMMON-SCHOOL ARITHMETIC. 

39 to 41. What will be the value of the following sums 
of money at the saine rate? 167 10s. 3Jd., 19 2s. 6d., 
10 10s. 10d. 

Answers: $95.63, $837.57, $52.72, $82.21. 



XXV. COMMON FRACTIONS. 
MULTIPLICATION. 

226. A fraction is multiplied by a whole number by 
simply multiplying the numerator without altering the 
denominator. [Art. 199.] 

| X 7=7X3, or ^-, which, reduced to a mixed number, 
equals 5J. 

227. Fractions can also be multiplied by dividing the 
denominator, without altering the numerator. [Art. 202.] 

AX5=f$=i, or If 

Multiply the following fractions: 

1. |X 5=1J 4. AXH=2-357 

2. |X 4=3 5. ^x 9=3.316 

3. |X 12=8 6. 7 8 T X 6=2.824 

228. Mixed numbers may be multiplied like compound 
numbers. 

7. Let it be required to multiply 4f by 7. 

Whole Nos. Eighths. 

Illustration. 4 5 EXPLANATION. Seven times 5 eighths 

7 equals 35 eighths, or 4 whole numbers 

~~^ ; o anc * 3 eighths. Seven times 4=28 and 

on* 4 make 32. Ans. 321. 

or 32f 

It will not be necessary for the pupil to write his work in so 
formal a manner as in this illustration. 



FRACTIONS. 217 

8. Multiply 6 by 12. 

6-J Multiplying 7 ninths by 12, we have 84 ninths or 9 
12 and 3 ninths. Then 12 times 6 are 72 and 9 make 81, 
.. 3 giving for the answer 81J. 

or 81] 

9. 6J X 8=?* 13. 35 1 X 9=? 17. 914 f Xl20=? 

10. 7| X 7=? 14. 60|-X12=-? 18. 63 X 15=? 

11. 8| X 6=? 15. 45 f X 8=? 19. 127 T \X 20=? 

12. 1,^X12=? 16. 64^X 6=? 20. 110 ^-X 14=? 
Answers: 54, 50f, 13^, 53 j, 109760, 952.5, 387.29, 

2543.636, 1542.3, 316, 730.5, 3S7-&, 365.3. 

229. To multiply one fraction by another, the numera- 
tors may be multiplied together for a new numerator and 
the denominators for a new denominator. 

21. f Xf=3r<V or f 

Here f is multiplied by 3, giving 6 thirds; but this 3 being sev- 
enths, 6 thirds is 7 times' too much. Multiplying the denominator 
[Art. 200] by 7 gives JL or ^. 

29 2\s 3\s 4\/ 5 1 20 1 
sX^Xs-X-g 3TTF 

These operations might have been abbreviated by what 
is called cancellation. In the first example, for instance, -| 
is to be multiplied by ^; that is, the numerator 2 is to be 
multiplied by the numerator 3 ; but the 2 is also to be di- 
vided by 3, for -| signifies that 2 is to be divided by 3; 
therefore, since the 2 is to be multiplied by 3 and divided 
by 3, it remains exactly the same, and the 3 of the de- 
nominator is said to cancel or make void the 3 of the nu- 
merator. In the following operations the canceled figures 
will be known by having a line drawn through them. 

* Many answers in this chapter are given in decimals carried to 
two, three, or four places, as in practice. Occasionally the plug 
sign is used to show that the decimals may be continued; at othei 
times the last figure will be found increased. 



218 NELSON'S COMMON-SCHOOL ARITHMETIC. 

2 $ 2 In the second operation, the 2 of the nu- 

I 7~7 merator and the 6 of the denominator are 

uncanceled, making f, which, reduced by 

_X-X-X- dividing both by 2, equals J. 

$^06 The 2 and 6 might have been canceled 

t $ $ a ^ s 5 ^ dividing both where they stood in 

-X-X-X- the question, as in the third example, plac- 

ing only 3 as a denominator, and 1 as a nu- 

merator. 1 is always to be understood 

where a number has been canceled. 



8 



23. i 

1 Some prefer arranging the terms of can- 

1$ 3 celing fractions as in the margin, with 

15 the denominator or divisor on the left, 

"^ and the numerator on the right. 

EXPLANATION. The first 2 was canceled in the 18, leaving 9; the 
24 and 9 were canceled by dividing both by 3, leaving 8 and 3; the 
74 was canceled by the second 2. 






24. iX -I X f =? 28. 1JX 

25. fX * XifX T V= ? 29. 

9K Is/ 8 N/ 5 \/ 7 ? 30 

^O. -^X T5"^34<^2T 

27. |XHXfXj=? 31. 

Answers: 3.375, 4.8, |, J, .05, 5.75, .0041, .0264+. 

230. Compound fractions are reduced to simple ones by 
multiplication. Let it be required to reduce \ of of | 
to a simple fraction. We know, by inspection, that one- 
half off is ^, and that \ of f is J 3 .the answer. 

By multiplication: 



By cancellation: -XTX-= 
?> I, Jl 



FRACTIONS. 219 

32. of | of ^=? 36. | of f of 9J =? 

33. * of # of 3*=? 37. fof }'<&& = ? 

34. f of % of -&=? 38. 4 of li of =? 

35. of Ij-of =? 39. iof |x*Xli=? 
Answers: 7.389, 1.75, .0068, .286, 1.5, -J, .4286, .3. 

231. To multiply one mixed number by another. 
The mixed numbers are reduced to improper fractions 
and multiplied as in Art. 229. 
40. 3J by 5*=? 



41.44X71=? 44. 4 T * r x3|=? 47. Ifx 8=? 

42. 2JX1=? 45. 2JX f=? 48. 72|x62>=? 

43. 3|X2 T 5 T =? 46. 6g- X2^=? 49. 87-5X16?=? 
Answers : 29f, 17^, l^J, 8||, 2.7, 1^, 17*i 4537.5, 

1458.3, 4.8, 14.583. 

50. At 11| cents a pound, what will 147J Ibs. of coffee 
cost? 

51. What will 7^ Ibs. of cheese cost, at 9J cents per 
pound? 

52. At 12J cents a pound, what will 120 Ibs. of sugar 
cost? 

53. What will 14^ Ibs. of beef cost, at 6j cents a pound? 

54. Fifteen and a half yards of muslin, at 9^ cents, will 
cost how much? 

Answers: 71 ts, $14.80, $16.59, 98 c, $1.43, $16.50. 

DIVISION. 

232. Division being the reverse of multiplication, to 
divide a fraction by a whole number, we divide the nu- 
merator or multiply the denominator. [Art. 2001.] 



220 NELSON'S COMMON-SCHOOL ARITHMETIC. 
2.. -j-g 4=? 5. --- 7=? 



;. ^^-9=?- 
. A-s-6=? 



6. ^.e^io^? 9. 

4. J8. 3? 7. I8_i_ 4? 10 " 7 - 9=? 

36 * ^Y * * ~~ 1 * 

Answers: ^, gL, ^ .1887, .076, .16, .16, .125, f. 

233. To divide mixed numbers. 
11. 21|-H-6==? 

Whole NOB. Fifths. EXPLANATION. 6 is contained in 21, 3 times 

'_ __ an d 3 ^ e ft" l n tne & f tne remainder there are 

3 3 15 fifths, which, added to the 3 fifths in the ques- 

or 3-| tion, make 18 fifths. % 6 in 18, 3 times. Ans. 3. 

12 1 2 ' 8 ? 

8)124- EXPLANATION. In this example we had 4 remainder, 

129 in which were 28 sevenths, and the one in the question 
& made 29. Then, as 8 would not divide 29 without a re- 
mainder, we multiplied it into the denominator, which made 56. 
Ans. 1||. 

13. 67f-r- 7=? 17. 167^25 = ? 21. 72f-- 9=? 

14. 445 _,_ 3== ? is. 21J--14=? 22. 148i-*-27=? 

16. 118^12=? 20. 22J-i-12=? 24. 175f-v-15=? 
Answers: llf 8^-, 5.5, 9f, 14||, 6.684, 1.523, 2.306, 
23.05, 11.7142+, 19||, 9, 1.861. 

234. To divide one fraction by another. 
OR i . 3 ? 

ft9 2~*~? 

1 . 3 1 V 4 4 ^ 2 

j-s-f =2 X -3 ^ B^- 

Multiplying the denominator of the dividend by 3, we 
have \ [Art. 200-1], but as the divisor is not 3 but 3 
fourths, we multiply the result by 4, giving |-, or 1^-; 
hence, we divide one fraction by another by inverting the 
terms of the divisor, and multiplying as in multiplication.* 

235. To divide by a mixed number. 

* CAUTION. The pupil will observe not to invert the terms of the 
number to be divided. 



FRACTIONS. 221 

26. 369--97rr=? 



Hero both terms are reduced to the same denomi- 
*4 obJ nation (thirds), and division performed as in whole 

~ numbers. 97J is contained in 369, 3|| times. 

^292 )H07(3f|i 
876 

231 

27. 76-f-2| = ? 30. 349 ~ 4J=? 33. 73^8 f =? 

128. 84-^-3|=? 31. 106f-7- 5|==? 34. 73 -=- ^=? 

^29. 82-r-7f=? 32. 276f-f-12 = ? 35. 191 1 -- T \=? 

Ans. 28fft, 23.05, 29.37, 10*f, 77.5, 18 J^, 8.565+, 
719^, 293/254. 

236. Complex fractions are unsolved questions in di- 
vision. 

36. 2J 7 

37. f-f-f =? 41. i of f -f-f =? 

38. f-^|=? 42. l|xiH-| =? 

39. ^-f- 1 = ? 43. 2J X T-4-| of |== ? 

40. i-H-i=? 44. 3^ * 

\/ . O rv-f- 7 f 

s\ ~^~ ' >? ^-*- "S"~~~ 

45. If 120-J Ibs. of cheese cost $14.80, what will 1 Ib. 
cost? 

46. Find the cost of 1 Ib. of coffee, when 15^ Ibs. 
cost $1.43. 

47. If 11^ yards of cassimere cost $16.59, what will one 
yard cost? 

48. If 9-J yards of muslin cost 71 cents, what will 1 
yard cost? 

Answers: 7^, 9^., 147/ 3 , 12^ r , 1.185, .6, .3, .53, .36, 
., 5$, 2.571. 



222 NELSON'S COMMON-SCHOOL ARITHMETIC. 



SUBTRACTION. 

237. To subtract fractions or mixed numbers from wlwle 
numbers. 

1. From 87 take 25f 

Whole Nos. Sevenths. TO subtract 3 sevenths, one whole number 

" is added to both terms. In 1 there are 7 

sevenths, from which subtract 3 and the re- 

61 4 mainder is 4. 1 to the 25 makes 26, which 

take from 87 and 61 is left, giving for the answer 61 A. 

The above formula being used for illustration only, the 
learner will be expected to write his operation as follows: 

87 
25* 

~61|~ 

2. 210371=? b. 1003819 * ==? 10. 2925 J =? 

3. 119 82-J? 7. 3785 10 -f- = ? 11. 167 89f = ? 

4. 61 4f = ? 8. 2168 14 f = ? 12. 3621^=? 

5. 54_ 51=? 9. 1765 777|=? 13. 218 36 1=? 
Answers: 987.4, 2153.571, 3774.14, 172.5, 36.875, 56. 

1, 48.875, 243.8, 172J, 3.25, 77.125, 181.625, 14|f, 3.75. 

238. To subtract a fraction from another of the same 
denomination. 

14. From take *. 

64=2 or I 
6 sevenths less 4 sevenths leaves 2 sevenths, the answer. 

15. |-i=? 18. |i-|| = ? 21. 

16.1-|=? 19. |_4 =? 22. 

17. i 3 2 - T V=? 20. ^- 3 \=? 23. 

Answers: J, II, .2, .25, .4, .14285+, ft, -i, .1, f 

239. ^b subtract one fraction from another, both terms 
must be of the same denomination, or reduced to the same 



FRACTIONS. 223 

denomination. | can be subtracted from f , but J can not 
be conveniently subtracted from | without first changing 
the denomination of one or both. 
24, Subtract J from J. 

EXPLANATION. The 6, being common to both fractions, is called 
the common denominator. To raise two fractions to a common de- 
nominator, both terms of each fraction may be multiplied by the 
denominator of the other. [Art. 197.] 

oc 3 2 9 ^?ft 4 1 ? Q"l 3 ___ 1 ^ 

rt/> 4^ i o on 2 5 y oo fj 1 9 

97 7 1 O Oft 7 4 > oo 2 1 ? 

Ail I TT 15" 4 OV. "If ?" * OO. 7j y -|-7j I 

Answers: |f, .2576, .3035+, .3, .07, ^, .375, .0396+, 
.075, .3096. 

ADDITION. 

240. Fractions of the same denomination are added to- 
gether by finding the sum of the numerators. 

1- l+f+l-2+5+7 thirds, or 4J. 

s' A+A+A+i-? 

Answers: 1.336, .851, {^J, J, 1 7 \, 2 T 4 T , 4 T \, 4|. 

241. Fractions of different denominations are added 
together by finding a common denominator, as in subtrac- 
tion, and proceeding as in last Art. 

8. Find the sum of f+f+f. 

3X5x6:=90. Common denominator. 

1 

\ 2X5X6= 60 Here the numerators 2, 3 and 5 are multi- 

'I 3X3X6= 54 plied successively by all the denominators but 

| 5X^X3= 75 their own. 2, for instance, is multiplied by 

-j oq &X6} the product of which is also the multi- 

I fpj . 189 Q ^ plier of the denominator 3, giving for the 

. 1S "^" -^TTT- fi rs t fraction |g instead of J. In the same 
way | becomes |J, and | Jg, making 189 ninetieths, or 2.1. 



224 NELSON'S COMMON-SCHOOL ARITHMETIC. 

When there are many fractions to be brought to the 
same denomination, it will be better to first divide the 
common denominator by each denominator. This gives 
the number which will raise both terms to the required 
denomination.* [Art. 197.] 

9- B- 4+ i=2 13. 8 +6 +12 =? 

10. H- 4+ i=? 14. 2$+ JH- if= ? 

11- !+ t+ 7T= ? I 5 ' Fr+ 9 + TT= ? 

12. 6J+7f+8=? 16. 6f +lf + 2 = ? 

Answers : 7J, 10.3714+, 1^, .4345+, 1.0707, 9.612+, 
32.25, 3.995+, 22.277, 26.75. 



19- 
20. 

21. | Of f +| Of 

22. |X T 9 ir+ of fX? of +l+5f +J=? 

23. 2|X6i+8+f of t+l of 3+7=? 

24. lX2+ of of 



*THE LEAST COMMON MULTIPLE 

Of several denominators is the least number which can be divided 
by them without a remainder. The following is the process for 
finding it: 

To find the least common multiple of 3, 4, 6, 8, 9, 12, 15. 

2)3 4 6 8 9 12 15 EXPLANATION. 2 was used as a di- 
3^3 2 3 4 9 6 HT visor of 4, 6, 8 and 12, and the. quo- 
2 77 ~ - ~ ~ ~ ~ tients set down. The other numbers 
were brought down and 3 divided into 



those divisible by it without a rernain- 



36 - der; and so the process was continued 
until no number could be found to divide the others without a re- 
mainder. The divisors and remaining numbers then being multi- 
plied together, produce 360 as the least common multiple. 

Like the operose method of finding the greatest common divisor, 
this is seldom used. 



FRACTIONS. 225 



25. 4f x5J+6iX2| of 

26. | of f +? of I of 

2T. 



28. 



29. f of f + _^ + 1 of ^ =1 

30. 4- + ^ of }f + | of 5^=? 

31. |f of7f of 9 + ~ of 14 ==? 

32. + |iof2f + |- of 6f=? 

33. ^of i|of9|+^of 8=? 

34. + + ii+e =^ 

Answers: 14^, 15/ T 8 3 +, 34.91+, 85.9146, .01269, 
7.14285, 26.1428, 5.25, 141.85, 175.85, 1.9642, 12 J, 
^W^ 3.4531+, 14.183+, 4.350+, 53.83, 
11.168+, 3.0278+, 4.0126+, 32.2678+, 11.694+, 
, 82.61. 



PRACTICAL QUESTIONS. 

1. In an invoice of goods there are the following items; 
required the amount. 

27 J doz. @ 9Jc 13f doz. @ 5^c. 

18 T 7 ^ " " 12 16| " 3J 

16^ " 12J 118^- 2J 

-4ws. $10.65. 

2. of a merchant's goods were destroye^ by fire, and 
what remained was worth $1637.50; what was his loss? 

3. A owns of a steamboat, B J and C the remainder, 
which is worth $10000; what is the value of the boat? 

4. J of a saw-mill belongs to A, to B, T 3 ^ to C, the 
remainder to D, and the profits for the year amounted to 
; $1680; what is each man's share? 

5. The par value of the pound sterling is $- 4 -; required 
the value of 1674 at 10% premium. 



226 NELSON'S COMMON-SCHOOL ARITHMETIC. 

6. A can do a piece of work in 8 days, B in 7 days 
and C in 6 days ; in what time can they do it if all work 
together? 

SOLUTION. A can do , B -f and G J of the work in a 
day. The sum of these fractions is -$%. If y^ can be 
done in a day, -J-JJ! (the whole) can be done in J^=2^-|-, 
or 2 days 3^ hours. 

7. There are 3 pumps placed in a coffer dam ; one will 
empty it in 10, another in 15 and a third in 20 hours; in 
what time can it be emptied by working all three at once? 

8. Express ^ of a day in hours, minutes, etc. 

SOLUTION. | of a day is the same as -| of 3 days. 

Days. Hours. Min. Sec. 

7)3 00 



10 17 8$ 

EXPLANATION. As 7 is not contained in 3 days, we reduce them 
to hours=72 hours, which, divided by 7=10 hours and 2 left, etc. 

9. In of a pound (British money), how many shil- 
lings and pence? 

10. In |- of a bushel, how many pecks, quarts, etc.? 

11. In \ of a ton (long weight), how many hundreds, 
etc.? 

12. Find | of 167 18s. 6d. 

13. Eeduce ^ an inch to the fraction of a foot. 

14. Reduce -^ a cent to the fraction of a dollar. 

15. What part of a pound Troy is \ an ounce? 

16. What part of a ton is J of a pound? 

17. | of a farthing is what part of a pound? 

18. | of f of $1600 is what part of $1000? 
Answers: $420; $210; $315; $735; $480.00: $3275; 

16s. Bd.; 3 pks. 4 qts.; 3 cwt. 1 qr. 9 Ibs. 5 oz.; 4 
hours; $8184; y^; ^ ; 3 ^ ; f; 3^; -g-^. 



PROPORTION. 227 



RATIO. 

242. The relation that one number bears to another is 
called ratio. The quotient arising from dividing one num- 
ber by another, of the same denomination, is the ratio 
between them. 

And as two quotients can be obtained from comparing 
any two numbers, it follow* that two ratios can also be 
obtained. The relation that 1 bears to 2 is J, and that 
which 2 bears to 1 is f . 

The sign of ratio is the colon. The above ratios would 
be expressed thus: 1 : 2 and 2 : 1, and would be read one 
is to two and two is to one. Some mathematicians divide 
the first term by the second; others, the second by the 
first. The former method is most used: 

6 : 3 will equal | or 2, f : ^=i=j, or 1|. 

i 

243. Numbers or quantities of different denominations 
can not have a ratio. We can not compare 3 trees with 
5 books. But if the numbers are capable of being re- 
duced to the same denomination, they can be compared ; 
for we can say 3 feet is to 2 inches, as it is the same as 
to say, 36 inches is to 2 inches. 



XXVI. PROPORTION. 

244. Two ratios may be equal to each other. 2 : 4~ 
:8. 

2 bears the same relation to 4 that 4 does to 8. 

245. When ratios are equal, the numbers or terms 
which compose them are said to be in proportion, and are 
written thus : 2 : 4 : : 3 : 6, and read 2 is to 4 as 3 is to 6. 



228 NELSON'S COMMON-SCHOOL ARITHMETIC. 

The first and last terms, as the 2 and 6, are called ex- 
tremes, and the second and third the means. 

246. The same ratio may arise by comparing 4 quan- 
tities, two of which are different in denomination from the 
other two. 

Tons. Tons. $ $ 

3 : 6 : : 6 : 12. 

The ratio is J, and if reversed, as 6 : 3 : : 12 : 6, it 
would be 2. 

247. If the extremes are multiplied together the pro- 
duct will be equal to the product of the means. 

3X12=36 
6x 6=36 

Hence, when any 3 terms are given, we can readily find 
the fourth by dividing the product by the odd term. If 
we had only the three first terms of the above proportion, 

Tons. Tons. $ 

that is, 3 : 6 : : 6, the fourth term would be found by 
dividing the product of 6X6, or $36, by 3=$12, or thei 
fourth term as above. 

To apply this in practice, we have only to suppose the 
3 tons and 6 tons to be coal, and the $6 the price of 3 
tons. Then 3 tons is to 6 tons, as the price of 3 tons is 
to the price of 6 tons. 

1. What will 35 Ibs. of sugar cost, if 7 Ibs. cost 77 c.? 

STATEMENT. 7 : 35 : : 77 is to the price of 35. 

7 : 35 : : 77 EXPLANATION. The means, 35 and 77, 

35 ing multiplied together, produce 2G95, and 

~^ r this divided by the given extreme, 7, gives 

~~.j the required extreme, 385, which must be of 

., the denomination of cents, in order that a 

7)2695 ratio exist between it and the third term 

~385 cents, ceuts - 
or $3.85. 



PROPORTION. 229 



The same by cancellation. 
5 



=385, or $3.85, Am. 

By placing our terms in fractional form, we have 35X77 for a 
numerator and 7 for a denominator. Then reducing both terms in 
the same ratio, the 7 cancels the 35, leaving 5X77 for a numerator 
and 1 for a denominator. 

2. If 27 Ibs. of butter cost $3.75, what will 16| Ibs. 
cost? 

3. Find the price of 12^- dozen of chickens, at 30 cents 
a pair. 

4. The price of 21 tons, 13 cwt., 3 qrs. and 15 Ibs. of 
hemp is $1680.55; what will 15 cwt. cost? 

5. What will 54 Ibs. 7-J- oz. cost, if 15 J Ibs. cost $8.47? 

6. If f of a ship cost $7000, what will T 9 a cost? 

7. If 6 men do a piece of work in 7 days, how long 
will it take 5 men to do it? 

STATEMENT. 5 : 6 : : 7 The 7 (days) having no ratio to the 

7 other numbers of the proportion, is 
5VL9~ placed to the right. At first sight, it 

' would seem that the proportion would 

8 1 be 6 : 5 : : 7; but 6 men do not bear 
the same ratio to 5 men that the time of the 6 bears to that of the 
6. A little reflection will convince the learner that 5 men would 

I require a longer time to do the work than 6 men, which would fail 
j to complete a proportion, as shown by the following 

j STATEMENT. The greater : the less : : the less : the greater, 
or, the greater X the greater=rthe less X the less! 

Hence, to find the second term of a proportion, it will 

jbe necessary to inquire whether more or less will be re- 

| quired. If more, put the greater of the two terms in the 

'second place; if less, put the less of the two terms in the 

second place. 



230 NELSON'S COMMON-SCHOOL ARITHMETIC 

8. If two men plow a field in 3 days, how long will it 
take 3 men to do it? 

9. If 26 yards of linen cost $13.50, what will 10 yards 
cost? 

10. If 3 coats can be made from 10J yards of cloth, 
how many can be made from 31^ yards? 

11. If the interest for $750 for 3 years, 4 months and 
10 days be $151.25 (360 days to the year), what is it for 
one year? 

12. The interest of 100, from 3d of April to 25th of 
February, is 6 5s. 9|^(Z. ; what is it per year? 

13. A, B and C are in partnership, and their gains for 
the year are $6757 ; what is each man's share, suppose A 
invested $1567, B 2600 and C 3798? 

The sum of their investments is to each man's invest- 
ment, as the total gains to each man's gain. 

14. M invests $6500, N $1487, $3654; in 3 months 
it is found that their gains are $1678; what is each man's 
share ? 

15. A lends B $1000 for 13 months 10 days; how long 
should B lend A $8271, to return the favor? 

16. If the shadow from a two-foot rule be 6 inches, 
what is the height of a tree that throws a shadow of 75 
feet? 

17. If 7 men can build 21 perches of masonry in a day, 
how long will it take 14 men to build 147 perches? 

Answers: $5.19, 9, 3, 7, 300, $1329.34, $936.89, 49, 
$5.90, $45, $4.50, $2.25^, $22.50, $7350, $58.09, $30.25, 
3,2. 

REMARK. This rule is of less utility to the business man or me- 
chanic than is generally claimed for it, as most of the problems can 
be solved in less time, and with fewer figures, by the application, 
of Multiplication and Division. Take, for instance, the 17th. If 



PROPORTION. 231 

21 perches can be built in a day, 147 can be built in 1477 days; 
and if 7 men can do it in 7 days, 14 men can do it in one-half of 
7, or 3J days 

COMPOUND PROPORTION. 

248. A proportion is said to be compound when it is 
composed of more than two ratios or four terms. 

1. If 3 men in 5 days, by working 8 hours a day, dig 
a cellar 15 feet long, 12 feet wide and 7 feet deep, in how 
many days will 2 men dig one 17 feet long, 14 feet wide 
and 6 feet deep, by working 10 hours a day. 

In this problem there are 11 terms and 5 ratios: the 
ratio between men and men, that between hours and hours; 
between feet and feet of the length ; feet and feet of the 
width, and feet and feet of the depth. In arranging these 
terms, we proceed as in simple proportion, Ex. 7, page 
229. 

Days. 

Men, 2 : 3 : : 5 1. Days are wanted; write days as 

Hours, 10:8 right-hand term. 

Length in ft., 15 : 17 2. Comparing men with men, we 

Width in ft, 12 : 14 find that it will take 2 men a longer 
Deptli in ft., 7 : 6 time to do the job than it took 3 men, 

BO we write the greater of the two terms (3) in the second place. 

3. Comparing hours with hours, we reason that it will take less 
time to do the job by working 10 than by working 8 hours a day, 
BO we write the smaller number on the right and under the second 
term. 

4. Comparing length with length, we reason that it will take a 
longer time to dig a cellar 17 feet long than it did to dig one 15 feefc 
long, so we write the greater (17) term under the second term. 

5. Comparing breadth with breadth, it will take a longer time to 
dig a cellar 14 feet wide than it did to dig one 12 feet wide, so we 
write the greater (14) under the second term. 

6. Comparing depth with depth, it will take less time to dig a 
cellar 6 feet deep than it did to dig one 7 feet deep, so we write tho 
smaller numbei under the second term. 



232 NELSON'S COMMON-SCHOOL ARITHMETIC. 
The same by cancellation. 



17x2 

=6$ or 6 days 8 hours.* 



$ 

$ 
17 



REMARKS. 1. The pupil should observe to have the terms of each 
ratio of the same denomination. 

2. The answer will be of the same denomination as the right- 
hand term. 

2. If 6 men in 15 days dig a trench 18 feet long, 7 
feet wide and 5 feet deep, in how many days will 21 meu 
dig a trench 125 feet long, 9 feet wide and 4 feet deep? 

3. What is the interest of $6784 for 2 years, 6 months 
and 15 days, at 6% per annum? 



STATEMENT. 



Days, 365 
Dollars, 100 



$6 
927 

6784 



Ans. $1033.77. 

4. The interest of $1467 for 3 years, 4 months and 12 
days is $450.72; what is the rate per cent.?f 

5. The interest on $786.55 at 10% is $176.44; what is 
the time? 

6. The interest for a certain sum of money for 4 years, 
2 months and 20 days, at 6%, is $100; required the prin- 
cipal? 

*In forming this proportion, we reasoned from what was given to 
what was required. For instance, in comparing men with men, we 
inquired if it would take 2 men a longer or a shorter time than the 
time (5 days) that it took 3 men to do it. 

fThe pupil can prove his own work by computing the interest by 
the method taught in the first part of this book. 



PARTNERSHIP* 233 



XXVII. PARTNERSHIP. 

249. WHEN two or more persons associate together to 
carry on a business, they are said to be in partnership, 
and are called a firm, house or company. 

Partnerships may be general or special. General part- 
nerships extend to the whole of the mutual dealings of 
the parties. Special partnerships are formed for some 
specific purpose, a single dealing or adventure. 

When more than two persons are engaged in business, 
it is usual to select the names of one or two of the mem- 
bers, with the term U 0o.," for the name of the partner- 
ship; as a business conducted by Messrs. Jones, Evans, 
Henderson and Norton might be called the firm or house 
of Jones & Co.* 

250. Each member of a firm becomes responsible for 
the acts and contracts of his copartners, in the way of 
sale, purchase, promise, agreement, etc., performed in the 
course of the usual business of tfce firm. If a partner 
draws a note or bill even in his own name, and it can be 
proven to be on account of the partnership business, he 

1 thereby renders the firm liable. 

251. An individual becomes a partner by allowing the 
community to presume that he is such, or by having his 
name appear on a sign or in a bill, card, etc. A secret 
partner becomes equally liable when discovered, as if his 
name appeared in the firm. 

252. A creditor of one of the members of a firm can 

*The name of a firm is not always derived from the members 
having the largest interest. Where precedence is not given to age, 
the names of the most influential are usually selected. 

20 



234 NELSON'S COMMON-SCHOOL ARITHMETIC. 

claim only the interest of the debtor in the partnership 
property after all claims against it have been settled. 

253. All partnership agreements should be written. 
The funds, property and merchandise furnished by 

partners for carrying on business are called stock or cap- 
ital, and the gains are called dividends. 

The liabilities of a partnership or individual business 
are the debts, and the assess, their available means, includ- 
ing the indebtedness of others to them. 

An inventory is a list or statement of those things which 
constitute assets. 

254. In keeping partnership accounts, each member of 
the firm should be credited with all that he brings into 
the concern or business, and be charged or debited with 
all that he takes out, just the same as if he had no inter- 
est in it. 

255. The calculations peculiar to partnership relate to 
the division of property and profits. 

1. A, B and C have been in business one year, and 
find they have made a net gain of $3476, which is to be 
divided as follows: A is- to have J, B J and C J; required 
the share of each. 

$Mp._$i738 j A's share; 8^-&=J869, B's share; and 
$869=C's share. 

2. X, Y and Z purchase a tract of land for $2000, X 
giving $600, Y $900 and Z the remainder. In one year 
afterward they sell it for $5500; required each person's 
share of the proceeds. 

3. A, B and C invest $2000 each. In three months 
their gross gains are $2000 ; expenses^ including $250 for 
additional services of C, $600: what will be each man's 
share of the gain? 



V . 



JOINT STOCK COMPANIES. 235 



XXVIII. JOINT STOCK COMPANIES. 
STOCKS. 

296. A Joint Stock Company is a body of men asso- 
ciated together in a species of partnership, to carry out 
some heavy undertaking requiring the investment of more 
capital than individuals or partnership companies com- 
monly possess. Joint stock companies are usually ineor-^ 
porated by act of legislature, with certain privileges. 
Railroads, canals, bridges, etc., are generally constructed 
by this species of combined interest, and many banking 
and insurance houses, scholastic institutions, etc., are 
owned and managed by joint stock companies. 

When an association of this kind is to be formed, a few 
leading persons make an estimate of the probable amount 
of capital required, divide it into equal shares of from 
$10 to $100 or $500, according to the nature of the un- 
dertaking, and issue certificates of ownership for each 
share. These are called certificates of stock, and are trans- 
ferable. Persons owning certificates are called stock- 
holders. 

Joint stock companies are usually managed by a presi- 
dent and board of directors^ elected for the purpose, by the 
stockholders. 

When shares sell for the price named in the certificate, 
the stock is said to be at par; if above this value, they 
are said to be above par; if below it, below par. 

Besides the stocks of companies, there are government 
stocks, which consist of bonds that have been issued by 
state officers, for the purpose of borrowing money. These 
draw interest at a specified rate. 



236 NELSON'S COMMON-SCHOOL ARITHMETIC. 

In dividing the profits of joint stock companies, it has 
been found more convenient to declare the dividend by 
percentage. 

1. What is the cost of 10 shares of railroad stock at 
5% below par, the original cost being $100 per share? 

Find the cost of 10 shares at $100 and deduct 5. 

2. A banking institution declares a dividend of 18% 
on a capital of $30000 ; what amount of money should a 
stockholder receive who holds 5 shares, valued at $200 
each? 

3. I hold 15 shares (each $100) of stock in gas-works, 
which have declared a dividend of 20%; how much am I 
entitled to after my gas bill of $20 is deducted? 

4. How many shares of United States stocks at 2% 
above par can I buy for $1224, the original cost being 
$100 per share? 

5. What amount of stock can I buy for $1683, if I am 
allowed 2% commission on the amount invested? 

The amount I am to receive is to be y-g-g- or ^ of the 
amount of stock purchased not -^ of $1683, for that 
would be commission on commission and investment. 

Let the amount to be invested be represented by |-g, 
and to this add ^j==|^, then we discover that $1683 is 
f-J- of the amount to be invested. 1 18 3 33=-^, or my 
commission, which, if we multiply by 50, will give us the 
amount to be spent, $1650. 

To prove this, find the commission on $1650 at 2%. 

6. A broker receives 1685, which he is desired to in- 
invest in State stocks; how much should he invest, and 
allow himself 2J% on the investment? 

7. What amount of stock can a broker buy for $16700, 
and allow himself J% on the investment? 

Answers: $180, $950, $280, $16658,35, $6619,51, 12. 



JOINT STOCK COMPANIES. 237 

257. When declaring dividends, it is customary to re- 
serve a part of the gains of business for current expenses. 
Such sum is carried to an account called a Contingent 
Fund or Contingent Expense Account. Dividends, in 
this way, are usually declared for an even rate per cent. 

8. A coal oil company, with a capital of $150000, 
gain! $31,493, and has concluded to declare a dividend of 
15%: how much will be left to apply to contingent fund? 

15% of $150000=$22500. 
$31493 $22500=$8993, Am. 

9. An insurance company gains $53,369.87, in six mos., 
on a capital of $500000, but does not consider it safe to 
declare a full dividend; how much will apply to contin- 
gent fund account after declaring a rate of 10% per an- 
num? 

10. What is the largest even dividend which can be 
declared by a company with a capital of $150000, whose 
gains are $13547.65? 

11. A stockholder, owning 20 shares, at $50 each, re- 
ceives a dividend of $120; what is the rate per cent.? 

12. The first dividend of a company is payable in bonds, 
by which a stockholder, owning 20 shares, obtains two 
shares worth $100 each; what was the rate of dividend 
declared? 

13. What would apply to contingent account where a 
dividend of 20% was declared on a gain of $316784.87, 
arid a capital of $2000000? 

Answers: $116784.87, 10%, 12^, 9%, $28369.87. 



238 NELSON'S COMMON-SCHOOL ARITHMETIC. 



XXIX. BANKRUPTCY INSOLVENCY. 

258. Bankruptcy signifies inability to pay. A person 
becomes bankrupt when he is obliged to give up his busi- 
ness for want of means to pay his debts, and to carry it 
on. Such an individual is said to have failed. Bank- 
ruptcy and insolvency are synonymous terms. 

Insolvent debtors usually transfer their property to 
other parties for the benefit of their creditors. This is 
called making an assignment, and prevents the individual 
debtors from recovering more than a share of the property 
apportioned to the amount of their claims. The person 
to whom an assignment is made is called an assignee, the 
property and claims of the debtor, his effects or assets, and 
his indebtedness, his liabilities. 

I. A person failing in business has the following effects 
to meet claims to the amount of $13000; how much 
should his creditors receive on the dollar? Merchandise 
to the amount of $3500, railroad stock to the amount of 
$2100 and personal claims to the amount of $1500. 

3500 
2100 
1500 

Amount of assets, 7100, which, reduced to cents, and di- 
vided by the amount of the liabilities 5 4 T 8 ^ cents, or 

*13|000)710|000(54 T 8 3 
65 



52_ 
~8~ 



IMPORTING. 239 

2. The amount of assets belonging to an insolvent 
debtor is $4684, and his liabilities $22000; how much can 
he pay on the dollar? 



XXX. IMPORTING. 

259. IMPORTING is the business of buying goods in a 
foreign to sell in a home market. A tax, under the name 
of duties or customs, is imposed by government on most 
imported articles of commerce. Such taxes are levied for 
the purpose of creating revenue to defray the expenses of 
government or to protect home manufactures and agricul- 
tural interests. Duties are regulated by a scale of prices 
called a tariff] and are altered according to the exigencies 
of the times or caprice of the administration. 

A liigli tariff signifies high rates of duties, and a low 
tariff, low rates of duties.* 

The persons appointed to examine imported goods and 
collect taxes are called custom-house officers, and their place 
of business, the custom-house. 

260. Duties are of two kinds: Ad valorem and Specific. 
Ad valorem duties consist of a rate per cent, on the value 
of goods as stated in the invoice; Specific duties, of a 
stated sum of money on the quantity imported, without re- 
gard to value, as $1 a gallon, $20 a ton. 

261. Certain allowances, called draft, tare, leakage and 
breakage, are made on goods charged with specific duties. 
These allowances sometimes consist of a percentage of the 
weight or quantity and sometimes of a specific deduction. 

Tare is an allowance made for the weight of the box, 

* Taxes are often levied upon exports as well as imports. 



240 NELSON'S COMMON-SCHOOL ARITHMETIC. 

barrel, bag, crate, etc., which contains the goods, and is 
usually calculated by percentage ; etc., after the deduction 
for draft is made. 

Draft or tret is an allowance made for loss by weighing 
in small quantities, and for impurities to which some 
goods are subject. 

On 112 Ibs., or less, it is 1 Ib. 

From 112 " to 224 Ibs., 2 Ibs. 

224 " " 336 " 3 " 

336 " " 1120 " 4 " 

1120 " " 2016 " 7 " 

More than 2016 " 9 " 

Leakage is an allowance of 2% on liquids in casks, 
paying duties by the gallon. 

Breakage is an allowance on bottled liquors, usually 
5%, but on ale, beer and porter, 10%. 

Gross Weight is the total weight of goods and box, bar- 
rel, etc. 

Net Weight is what remains after all deductions are 
made. 

We shall not trouble the learner to work out any ques- 
tions in this chapter, as it rarely happens that young peo- 
ple have them to do in business. 



XXXI. FOREIGN EXCHANGE. 

262. IN calculating Foreign Exchange, the money of 
one country has to be represented in that of another. A 
bill drawn in New York on a merchant in England will 
be expressed in pounds, shillings and pence. 

The draft, though not stated in the question, is to be deducted bo- 
fore other allowances are made. 



FOREIGN EXCHANGE. 241 

263. The relative value of moneys of different coun- 
tries depends on the par of exchange and the course of 
exchange. 

264. The par of exchange is the comparative value of 
the coins of the different countries, and is fixed, while the 
relative purity of the coins is the same. The par of ex- 
change between the United States and Great Britain is 
$4.86 to the pound sterling. Formerly, when the silver 
of the United States was purer, the par value of the 
pound sterling was 4|, and exchange is still quoted from 
this par, or 9^ premium on the old currency being the 
present par value. 

265. The course of exchange usually depends upon the 
relative state of indebtedness of the merchants of the dif- 
ferent countries, and the supply of gold and silver; ac- 
cordingly, the course of exchange will sometime be above 
and sometimes below par. 

EXCHANGE WITH GREAT BRITAIN. 

FORM OP A FOREIGN BILL. 

Exchange for 1567. CINCINNATI, June 3, 1867. 

Thirty days after sight of this first of Exchange, (sec- 
ond and third of the same tenor and date unpaid,) pay to 
the order of William Tuechter, the sum of One thousand 
five hundred and sixty-seven pounds sterling, value re- 
ceived, and place to my account as advised.* 

To William Morgan, JSsq., J. B. TREVOR. 

Liverpool, England. 

* Foreign bills are generally drawn in sets of two, three or four; 
that is, copies of the same bill are made out and transmitted by 
different conveyances to the payer, one of which being received and 
accepted, or paid, the others to be void. These copies are called 
First, Second or Third of Exchange. The above is a copy of the first. 
The others are drawn in a similar manner. 



242 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



BRITISH OR STERLING MONEY REDUCED TO 
FEDERAL MONEY. 

GIVING the cost of British money is called quoting it, 
and the rate, the quotation. 

Sometimes the total cost of a pound is given; at other 
times, only the premium on the old par. 

266. When the Amount is given in the Quotation. 

1. What is the value of 157 9 2 in Federal money @ 
$4.86 to the pound sterling? 

SOLUTION. Since 1 is equal to $4.86, 157 will be 
equal to 157 times 486, or $763.02. Then taking aliquot 
parts of a pound, we have 6 shillings and 8 pence:=J, 
and 2 shillings and 6 pence^^J of a ; add J and -J of 
$4.86 to $763.02=765.25. 

TABLE OF ALIQUOT PARTS 



Of a Pound. 


G>/ a Pound. 


0/ a Shilling 


s. d. 


d. 


d. 


10 is i 


10 is 7 V 


6 is 


6 8 




8 


i 
sir 


4 


4 


5 


I 


7J 




3 


i 


4 




6 


5 


2 




3 4 





5 


A 


I* 


^ 


2 6 


i 


4 




1 


T 


2 




3 


A 


Of 




1 8 


_C 


2 




0^ 


^4 


1 4 


IT 


1J 


yi(5 


< 




1 3 


A 


1 


T0 




1 


7>Tf 







2 to 4. At $4.84, what will the following sums amount 
to? 345 14 0, 15 3 4, 365 7 6. 

5 to 7. 147 13 4 @ $4.87, 425 17 6 @ $4.44, 652 
10 @ $5=? Answers: $5872.53, $3515.02 

267. 2b reduce British to Federal money when the Pre* 
mium only is given. 



FOREIGN EXCHANGE. 243 

8. What is the value of 221 15 6 @ 8% premium? 
The premium is reckoned on the old par value; that is, 
$4.44| to the pound, or $40 to 9. 

12)6.0 Reducing 15 shillings and 6 pence to the decimal of a 
P oun< *j we have -775. 



775 

Hence, 221 15 6=r-221.775, which, multiplied by 40 
and divided by 9=$985.666, to which, if we add 8%, we 
shall have for the answer, $1064.52. 

What is the value of the following in Federal money? 

9. 1424 19 9 @ 7J% 13. 313 8 4 @ S%% 

10. 3575 18 6 @ 8 % 14. 505 19 6 @ 9 % 

11. 1100 12 6 @ 8J% 15. 3737 12 3 

12. 111 @ 9f # 16. 649 4 6 

Total, $29793.55. Total, $25270.28. 

267. When gold is at a premium, the rate may be 
added after the rate of exchange. 

Take the 8th example, assuming gold to be at a pre- 
mium of 45 cents, or 45%. 

Value of 221 15 6 at the old par^$985.666. 

To this add 8% premium on British money and 45% 
on gold to the sum, and we have $1543.56. 

268. TO REDUCE FEDERAL TO BRITISH OR 
STERLING MONEY. 

CASE I. 

When the Amount is given. 

17. At $4.87 to the pound sterling, what will be the 
value of $37654? 

Since $4.87=1, $37654 will be the equal to as many 
pounds as $4.87 is contained times in that number. 
W65400-r-487==7731.827 f which reduced=7731 16 6J. 



244 KELSON'S COMMON-SCHOOL ARITHMETIC. 

Reduce the following to British money: 

18. $3674.87 @ $4.87. 19. $67845.18 @ 4.44|. 

Total, 16019 15 2. 

CASE II. 

WJien the Premium only is given. 

20. At 9% premium on sterling money, what will be 
the value of $3964? 

At 9% premium $1.09 is worth only $1; therefore, 
$3964 are worth as many dollars as $1.09 is contained 
times in it. 

396400^-109=3636.697, which, reduced to pounds by 
multiplying by 9 and dividing by 40=818.2568, or 
818 5 1J. 

21 and 22. Reduce the following to British money : 
$3165 @ \%, $1678.90 @ 9%. Total, 996 2 2J, 



XXXII. DTSITRAETCE. 

269. INSURANCE is a guarantee against loss. It may 
be of several kinds, as Fire, Marine and Life insurance. 

270. Insurance on fixed property is called fire insur- 
ance; that on movable property, as goods in course of 
transportation, ships, etc., is called marine insurance; that 
which guarantees the payment of a sum of money to a 
survivor at the death of an individual, life insurance. 

271. The act of insuring is termed taking a risk; the 
amount paid for insuring, the premium; arid the paper 
upon which the contract is written, the policy. 

272. When the risk is heavy, the insurer sometimes 
re-insures in another company. 

273. In time of war, the rates of insurance increase 



INSURANCE. 245 

with the danger to which the property is exposed, or else 
the company secures itself by inserting in the policy ex- 
ceptional matter called the war clause. 

The rates of insurance vary according to the exposure 
of the property and the character of the property itself; 
the greater the risk, the higher the rate. 

Insurance can be obtained from one day to a term of 
years, giving a range of rates, from a small fraction of one 
per cent, to three, four and even higher rates per cent. 

1. How much should be paid to insure a house valued 
at $1674, premium being IT%, and policy $1.50? 

2. At 2^% premium, what should I pay on $6710 
worth of goods? 

3. At 4% premium, what should I pay on machinery 
and material in a factory, the estimated value of which is 
$6600? 

4. A company takes a risk of $35000 in a block of 
buildings, at l-g-%, and re-insures $15000 in another com- 
pany at l|r%; how much premium does it realize? 

5. At y 1 ^ of \% for 10 days, what should I pay on 
$30000? 

6. I have insured $16000 for 3 months at T 5 a of \%\ 
how much should I pay? 

7. What will be the insurance on merchandise worth 
675, to be shipped from Liverpool to New York, at 4: 
guineas per cent.? 

Answers: 28 7s., $26.61, $167.75, 30.5, $80, $30, 
$264.00, $337.50. 



246 NELSON'S COMMON-SCHOOL ARITHMETIC. 



XXXIII. DUODECIMALS. 

274. DUODECIMALS, like decimals, is a species of cal- 
culation which enables the operator to compute fractional 
quantities as whole numbers. 

12"" fourths make 1 third. 
12'" thirds make 1 second. 
12" seconds make 1 prime, or inch. 
12' primes or inches make 1 foot. 
1 inch is the fa of a foot. 

1 second is the j 1 ^ of an inch, or y^ of a foot. 
1 third is the fa of a second, jfa-g of a foot. 
1 fourth is the fa of a third, or 2 ^-^g- of a foot. 
As applied to mechanical pursuits, duodecimals have 
seldom to be subtracted or divided; hence, the following 
exercises will be confined to multiplication and addition 
exclusively. 

1. Multiply 2 ft. 5 in. by 3 ft. 4 in 

25 ! Writing the dimensions as in the margin, we com- 

3 4 mence with the left-hand figure (3 feet), and say, 3 

^ jr times 5 inches are 15 inches, or 1 foot 3 inches; write 

Q Q 3 in the inches' place. 

2. Then 3 times 2 are 6, and the 1 foot carried makes 

808 7 feet, which we write in the place for feet. 

3. We next multiply by the 4 inches; that is, we multiply 5 inches 
or y 5 of a foot by 4 inches, or - 4 ^ of a foot. The result will be -f-fr, 
but to avoid fractions, we call the result 20" or 1 inch 8". Write 
8 inches to the right of and below the 3 inches. 

4. Then 4 times 2 are 8', and !'=$' or inches, which we write in 
the inches' place. 

6. We now proceed to add them. There being nothing to add to 
8", we set it down ; then 9' and 3' are 12' or inches=l foot. Write 
and add 1 to the 7, makes 8 feet: the product of 2 ft. 5 in.X3 ft. 4 
in.=:8 0' 8'. 



DUODECIMALS. 247 

ANOTHER WAY. 

2. Multiply the following dimensions together : 10 ft. 
7 in.X3 ft. 8 in.x7 ft. 9 in. 

10 .7 Here we commence to multiply by 

3 8 the left-hand figure (3), and write the 

~~3fi 9? result without reducing to a higher 

denomination. 3X10 ft=30 ft., and 
7 in.X3=21 in. Then multiplying by 



30 101 56 1st pro. the 8, we write the first product un- 
7 9 der itself as the multiplier, and the 



210 707 392 second product, 56, one place further 

270 909 504 ^ ^ ie r *&kt. Adding these, we have 

the product of two divisors. 

Proceeding in the same way with 

the 7 and 9 of the third dimension, we add together the products 
and reduce them to higher denominations, by which we get 300 ft. 
8' 11", or 300 J* 2 ft.+Jj1 =300J ft., nearly * 

ft. in. ft. in. ft. in. ft. in. ft. in. 

3. 17 IX 3 4=? 7. 4 8X6 4x17 2=? 

4. 14 6X 7 8=? 8. 3 9X2 6x11 0=? 

5. 21 9X14 11-=? 9. 21 11X6 7x17 8=? 

6. 18 8X16 7= 

Total answers : 802 1' 3" and 3159 6' 3" 8'". 

ft. in. ft. in. ft. in. ft. in. ft. in. 

10. 21 7XH 10=-? 13. 31 7X3 2x3 3==? 

11. 13 9x17 4=? 14. 26 3x9 5x7 7=? 

12. 33 7X29 3=? 15. 17 1x6 7x0 9=? 
Total answers: 1476 7", 2283 10' 9" 6'". 

Mechanics preferring common or decimal fractions to 
duodecimals, seldom use the latter. 

The following example is worked by both methods: 

*For this very simple method, we are indebted to J, C. Kinney, 
Esq., of Reading, Ohio, not having seen it before. Its simplicity 
would suggest it as the best method to teach this otherwise difficult 
rule. 



248 NELSON'S COMMON-SCHOOL ARITHMETIC. 

16. How many squares of flooring in 3 rooms measuring 
18 ft. G in.Xby 15 ft. 8 in., and what is the cost of lay- 
ing, at 50 cents per square? 

-X^Xi=869 ft., or 8.695 sqs. 

18 6 
15 8 



277 6 
12 4 



289 10 
3 

869 6, or 869J sq. ft., which, reduced to 
squares of 100 feetr=8.695 squares. 8.695x50 cents 
4.347, or $4.35. 

17. What is the cost of laying 4 floors of the following 
dimensions, at 75 cents per square? 18 ft. 9 in.Xl7 ft. 3 in. 

18. What will be the cost of shingling a roof which 
measures 53 ft. 6 in. long, and 5 ft. 8. in. from the ridge 
to the outer edge of the wall, at $1.50 per square? 

19. The average breadth of a board is 1 ft. 4 in., and 
the length 23 ft. 9 in.; what number of feet does it con- 
tain? 

20. How many solid feet in a log measuring as follows? 
45 ft. 4 in.Xl ft. 6 in.Xl ft- 3 in. 

Answers: $9.09, 85 ft., $9.70, 31| sq. ft. 



XXXIV. INVOLUTION EVOLUTION. 

275. THE process of multiplying a number by itself a 
certain number of times is called Involution, while that of 
finding the number thus raised, or the reverse process, is 
called Evolution. 



IN VOLUTION EVOLUTION. 949 

276. A number multiplied upon itself is raised to the 
power; the second power multiplied by the number 

is raised to the third power; the number of the power be- 
ing indicated by the number of times the original number 
has been used. 

The second power is also called the square, because the 
number of square feet, inches, etc., is found by multiply- 
ing the number contained in one side by itself. For a 
similar reason the third power is called the cube. 

277. The power of a number is indicated by a small 
figure over the right of the number, thus : 5 3 , which shows 
that the third power of 5 is understood. This figure is 
called the index or exponent. 

278. Decimals are raised to any power in the same way 
as whole numbers, with the difference of placing the deci- 
mal point, while common fractions are involved by multi- 
plying the numerators and denominators separately. The 
second power of .5 is .5X-5 or -25, and the second power 

offis|x|= ? V 

279- The product of two numbers can not consist of 
more figures than there are in the two factors, and can 
consist of only one less than in the two factors. Take 9, the 
largest number of a single figure, and multiplied by itself, 
it produces only two figures, 81 ; and take 10, the small- 
est number of two figures, and multiply it by itself, and 
it produces three figures. This principle applies in find- 
ing the roots of numbers. 

1. Square the following numbers: 3, 7, 9, 4, 6, 15, 27, 
89, 97, 112. 

2. Raise the following numbers to the powers indicated : 
3 s 9 5 , 26*, 30 5 , 87*, 250 3 , 189 3 . 

280. The number from which any power is raised is 
called -the root of that power, and the process of finding 
that number is called extracting the root. 



250 NELSON'S COMMON-SCHOOL ARITHMETIC. 

281. The root of a number derives its name from the 
exponent of the power, the second or square root being 
from the second power, the third or cube root from the 
third power; and is indicated thus: -j/, square root; ^K, 
cube or third root; j^, the fourth root. 

282. A complete power is one which can have its root 
extracted. A surd is one which can not have its root 
extracted. 4 is a complete power, while 5 is a surd. 

TABLE. 

THE SQUARE ROOT OP 

1=1 36= 6 121=11 256=16 

4=2 49= 7 144=12 289=17 

9=3 64= 8 169=13 324=18 

16=4 81= 9 196=14 361=19 

25=5 100=10 225=15 400=20 

283. Since the product of any two numbers can not 
consist of more than four or less than three digits or fig- 
ures, the number of which a root is composed can be 
found by separating the squares into periods of two figures 
each, thus : 256 indicates that two figures formed the root, 
2,56 being the periods, the root of which is 16; and in 
the same way 746496 is pointed, 74,64,96, indicating that 
three figures composed the root. The square root is 864. 

THE EXTRACTION OF THE SQUARE ROOT. 

1. The square root of 765625 is how much? 

76,56,25(875 

64 1. Commencing at the right-hand, the power is 

17 M 9"R separated into periods of two. 

7 1 TfiQ ^* ^ e nearest S( l uare root of the last period is 

then taken, which gives 8, or 800. [Art. 83.] 

1745 )8725 Writing the 8 in the quotient, and squaring the 
8725 number, we have 64 (6400), which is written un- 
der the 76, or 76,00,00 



INVOLUTION EVOLUTION. 



251 



3. Subtracting this 64 from the 76, we have a remainder of 12, to 
which another period (56) is annexed, making 1256. 

4. For a part of the new divisor, the 8 of the quotient is doubled, 
giving 16 as a trial divisor. Finding it is contained 7 times, (the 
7 being included in the divisor,) that figure is annexed, making 
167, and the product is completed. 

5. 1169 being subtracted from 1256, leaves 87. 

6. Annexing the last two figures of the dividend, the last figure 
of the divisor is doubled, as before, making 174. This number, 
with the last figure of the quotient (5) annexed, is contained in 
8725, 5 times without a remainder, making the square root 875. 

RECAPITULATION. Separating the power into periods, we find the 
highest root of the last period. This we place as the first quotient 
figure, and subtract its square from the period. To the remainder 
annex the next period, and for a trial divisor double the last figure 
of the divisor. To this divisor we annex the next quotient figure 
and multiply as in long division. To the next remainder is an- 
nexed the next period, and to the last divisor is added its last 
figure, which is the same as to double the quotient, and the opera- 
tion proceeds as before. 

2. -t/1683129 is how much? 



1,68,31,29(1297.354 



22 

2 



249 
9 



2587 
7 

25943~ 
3 



259465 
5 



2594704 



68 
44 



2431 
2241 



19029 
18109 



92000 

77829 



1417100 
1297325 



NOTE 1. This answer may be 
carried out to any number of places 
by annexing ciphers, as has been 
done to produce the .354 of the quo- 
tient. Three figures, however, are 
sufficiently correct for practical 
purposes. 

2. To find the square root of a 
decimal quantity, we commence at 
the left to point off the periods of 
two figures. 146.739 would be 
pointed thus: 1,46.73,90. 



11977500 
10378816 



252 NELSON'S COMMON-SCHOOL ARITHMETIC. 

The pupil can prove the accuracy of his calculations by 
squaring the root obtained. 

3. v/14161. 8. j/16820.17. 



4. 1/625. 9. 1/23467.809. 

5. !/ 99980001. 10. ^167037^827 



6. j/99999.8000001. 11. T /456789.375. 

7. y/7837619. 12. |/10963.849. 

284. The square root of a fractional number is found 
by extracting the root of each term. The square root of 

sV : =V / 2 4 7= 1- 

285. Decimals are pointed off in periods from the right. 
.31671 is pointed thus: .31,67,10. 

286. The square root of the product of two numbers 
gives a mean proportional between them. |/5x20= 
1/10010, the mean proportional between 5 and 20. 

287. The square root of the area of a square is equal 
to the length of the side.* 

13 to 17. Find the mean proportional between 7 and 
175, 121 and 36, 6 and 24, 42 and 38, 16 and 49. 
Answers: 35, 28, 66, 39.949+, 12. 

18. A square garden contains 2916 yards; what is the 
length of a side in feet? 

19. A pavement is 112 feet long and 7 feet broad; 
what will be the length of the side of a square of equal 
area? 

20. How many yards of ground in the side of a square 
which would be equal to a lot measuring 144x196? 

21. What is the length of the side of a square piece of 
land which contains 25600 acres? 

Answers: 28, 168, 162, 160, 2023.85+. 

*The area is the contents of the surface, or the number obtained 
by squaring the side of a square. 




INVOLUTION EVOLUTION. 253 

288* The surfaces of circles are to each other as the 
squares of their diameters or circumferences. 

289. A triangle is a figure having 
three sides and three angles, or corners. 
If one of these angles is square, it is 
called right angle and the triangle is 
called a right-angled triangle. 

290. Two square figures, one having 
each of its sides equal to the perpendic- 
ular, and the other having each of its 

sides equal to the base, will, together, be equal to a square, 
each of whose sides is the same length of the hypotheuuse.* 
Let the perpendicular be 3, the base 4 and the hypoth- 
enuse 5. 

Then 3X3= 9, the area of the first square. 
4X4=16, " " " " second " 
25, " " " " third " 

But 25 is the hypothenuse squared or multiplied by it- 
self; therefore the square root of the square 25 will be the 
length of the hypothenuse. The square root of 25=5. 

Hence, the square root of the sum of the squares of the 
base and perpendicular will give the hypothenuse. 

And the square root of the difference ,of the squares of 
the hypothenuse and either of the two sides will give the 
third side. 

22. What length of a ladder will reach across a 15-foot 
alley to the top of a house 30 feet high? 



1125, the square root of which is 33 ft. 6J t?i., nearly. 

*The pupil should construct a diagram, with these squares 
his slate. 



254 NELSON'S COMMON-SCHOOL ARITHMETIC. 

23. What is the diagonal of a room 18 feet by 16? 

24. A ladder 30 ft. long, placed between two trees, 
reaches to the height of 27 feet on one of them and 25 
on the other; what is the distance between them? 



XXXV. EXTRACTION OF THE CUBE ROOT. 

291. THE cube root of a number is such a number 
which, if multiplied upon its square, will make that num- 
ber : 2 2 X2, or 2 3 r=8. The cube root of 8=2. The sign 
of the cube root is -j*K. 

The cube root of any number consisting of three figures 
will be a number represented by one figure ; the cube root 
of a number containing more than three and less than 
seven figures, will be one containing two figures ; hence, 
we point off the figures by threes instead of twos, as iu 
square root. N 

1. Find the cube root of 262144. 

262 144(64 EXPLANATION. The near- 
g3 216 est cube root of the first be- 



10800 ' placed to the left and its 

6 x30- 79? cube taken ' which is 216 

P X*A^ (216000). This subtracted 

. _ from the dividend, leaves 

11536 46144 46, to which is annexed the 

next period (144), making 
46144. 

For a trial divisor, 6 3 is then multiplied by 300, giving 10800, 
which is contained in 46144, 4 times. 

The part of the root previously obtained is multiplied by this, 
and that product by 30, giving 720. The square of the last, figure 
of the root is then taken, and the three results added, making 11536, 
which, multiplied by the last figure in the quotient, gives 46144. 






EXTRACTION OF CUBE ROOT. 255 

2. Find the cube root of 17596287801. 

17 V 596 V 287 5 801(2601 
2*= 8 

2 2 X300 =1200 )9596 
2 X<>X30= 360 
6 2 = 36 

1596 9576 



20287801 

260 2 X300 =20280000 
260X1X30= 7800 
I 2 = _1 

20287801 20287801 

This differs from the last example only in the cipher of the quo- 
tient. 

Finding the trial divisor (26 2 X300) was not contained in the new 
dividend, a cipher was annexed to the quotient, and another to the 
trial divisor, giving 260 2 X300 or 20280000. 

This being contained in the dividend 1 time, the former part of the 
quotient was multiplied by it and 30, and with the square of the 
last figure (1) added to the trial divisor, as before, giving 20287801, 
which, multiplied by 1, completed the extraction. The cube root 
is 2601. 

RULE. 

Find the greatest root of the left period, place it in the quo- 
tient and divisor , and subtract its cube from the dividend. 

To the remainder annex the next period, and, for a trial 
divisor, multiply the square of the root thus obtained by 300. 

Divide the new dividend by this divisor, and enter the 
product of it, with the root already obtained and 30, under 
the divisor ; under this enter the square of the last quotient 
fyure, and the sum of the three numbers will be the true 
divisor. 

Multiply this divisor by the last quotient fyure, and s?/7>- 
tract the product from the dividend; to the remainder annex 
another period, and proceed as before. 



25G NELSON'S COMMON-SCHOOL ARITHMETIC. 

3 to 7. -^389017=? ^259696072=? ^5735339=? 
^219365327791=? f 99252847= ? 

APPLICATION OF CUBE ROOT. 

292. A cube is a solid body, having all its sides of 
equal length. Any two sides of a cube multiplied to- 
gether will give the superficial contents of one of the faces 
of the cube, and this multiplied by another side, will give 
the solid contents; therefore, the cube root of the number 
of feet, yards, etc.. contained iu any solid, will give the 
side of a cube of equal bulk. 

7. An irregular block of stone contains 15781 cubic 
feet and 1333 cubic inches, or 27270901 cubic inches; 
what will be the side of a cube of equal solidity? 

8. Required the depth of a cubic cistern that will con- 
tain 3375 feet. 

9. What will be the side of a cubic bin or box that 
will contain 20 bushels of wheat? 

Answers: 15 ft.; 35.03 in.; 301 in. or 25 ft. 1 in 



XXXVI. AEITHMETIC APPLIED TO THE 
TRADES, FARMING, ETC. 

CARPENTRY. 

THE Carpenter proper may be called the outside and the 
joiner the inside carpenter. The distinction, is seldom ob- 
served. 

Master Carpenters are sometimes called Builders. They 
will contract for the entire work of an edifice, and super- 
intend its construction. The legitimate business of the 
carpenter is to prepare and fit all the wood-work used in 



CARPENTRY. 257 

building houses. His prices depend on the quality of ma- 
terial and style of finish. 

Plain work on one side of white pine lumber is taken as the 

unit of measurement, and is called 1 

Plain work on poplar is 1J 

Plain work on ash, oak, etc 2.} 

Plain work on maple 3 

Segmentat or Norman work on white pine 2J 

Gothic work on white pine 3 

Serpentine, or the Oriental variety, plain 5 

Domes 9 

Floors, roofs, partitions and weatherboarding are meas- 
ured and charged for by the square 100 sq. ft. 

The quantity of good lumber required for a square of 
flooring is 112 feet, -J- or 12 J% being allowed for waste. 

The quantity of good pine shingles required for a square 
of roofing is 1000. 

USEFUL HINTS ON BUILDING. 

Persons about to erect buildings in cities, should obtain a permit 
from the Board of Improvement, else they subject themselves to 
damages for placing obstructions in the street. 

Proprietors of adjoining lots and buildings should be notified of 
the digging of cellars, or the making of other excavations that 
would endanger their property. In Cincinnati, cellars may be dug 
12 feet deep without, the risk of incurring damage. To excavate or 
build cellars deeper than this, or to build sub-cellars, permission 
should be obtained from adjoining proprietors. 

Where property is valuable, it will be to the advantage of persons 
who purpose building, to remember that vaults, cellars, cisterns, etc., 
may be said to occupy no room; and that license may be obtained 
from the Board of Improvement to extend cellars or vaults under 
the sidewalk or pavement. 

Foundation walls should be made thicker than the main walls of 
buildings, and the latter, to be secure, should rest on the middle 
of the foundation, allowing it to project on both sides. 



258 



NELSON'S COMMON-SCHOOL ARITHMETIC. 



PRICES OF CARPENTER WORK, 

WITH C10LU AT PAR, FOR A DWELLING-HOUSE OF TEN ROOMS. 



Cellar windows, usual size, 

with sash $1.50 

Cellar steps, good 2.00 

" doors, each ,... 4.00 

First floor joists, IJin., per 

square,, , 0.75 

Floors, per square 0.65 

Trimmers, per foot 0.10 

pinning, " " 0.02 

Roofing, per square 1.50 

Hip rafte*, per foot 0.06 

Valley " " " 0.06 

Cants 0.01 

" 0.03 

Trap-doors, each 1.50 

Ceiling joists, per square... 0.50 
Partitions, " ... 0.35 
Poor heads, " ... 0.35 
Inside door-frames, per ft.. 0,04 
Outside " " ".. 0.05 

Window frames, plain 0.04 

Beads for do., soft wood.... 0.01 
" " '* hard " .... 0.01 J 

Box. W. frames, soft 0.08 

" hard 0.11 

" " beads 0.01 J 

R. sills, ,.. 0.06 

M. rails..,.! 0.06 

Betting frames .,,,,.,.,.,. 0.07 

Pocket and pullies, perpr.. 0.35 
Hanging sashes, per pair.. 0.20 
Base, 6 in. wide, plain, per 

lineal foot ,..,.. 0,05 

JJase mold..,..,.... 0.08 

<< " large. ,. 0.09 

Casings, 6 in 0.05 

" 5 ..,., 0.04 

Plinths, each., , 0.10 

Caps, 7 in., per foot ,0.10 

" 8 " '< " 0.08 

" 10 " M 0.10 

Cornice 0.25 

^helves, conimon,.,.,. , 0.03 



Shelves, best $0.05 

Cloak rails. 0.03 

Trimming doors 0.30 

Door sills 0.20 

Mantels, each 5.00 

" common, each...< ( 4.00 
Cupboard front, per sq. ft.. 0.04 

Window stools, per foot 0.06 

" " 0.18 

" large 0.22 

Cornices. 
Gutters on eave inverted, 

per lineal foot 0.13 

Do., mold 0.18 

Do., 3 members 0.23 

Do., 4 " 0.25 

Do., mold 0.37 

Do., " brackets 0.40 

Do., 6 members 0.30 

Do,, 6 modillions, etc 0.42 

Do., truss, each $3 to 5.00 

Truss usually referred to 
measurer. 

Fastening ornaments 0.04 

Bracket cornices 0.05 

Lining from $0,04 to 0.05 

Tubes 0.25 

Porticoes. 
Square colums, 8 to 9 in., 

per foot 0.15 

Capitals, each 0.75 

Best 1.25 

Porch, front, per foot 0.15 

" panel 0.20 

" extra 0.35 

'< cornice, plain 0.30 

" full..,,,, 0.50 

Ornamental left to meas- 
urer. 
Floors and roofs, per sq. ft. 0.25 

Framework 0.20 

Sills O.tt 



MASONS' WORK. 259 

In carpenter work, the cost of material is about equal 
to the cost of labor. For calculations, see duodecimals, 
page 246. 

MASONS' WORK. 

293. A bill of prices and a standard of measurement 
are generally fixed upon by mechanics of the various cities 
of the Union, and contract work is charged for at a cer- 
tain rate per cent, on the bill, according to agreement. 

Master masons, carpenters, etc., usually select one of 
their number who is expert in figures, and a good judge 
of work, to attend to the measurement of work done. 
This person is called a measurer, and his fee is paid one- 
half by the workmen and the other by the employer. The 
following rules are taken from the Cincinnati Stone-ma- 
sons' bill of prices: 

"RULES FOR MEASURING. 

"1. All work is to be measured by the perch of 24J 
cubic feet. 

" 2. All work to be measured from each outside corner, 
including all openings under eight feet wide. 

"3. All openings less than five feet wide to measure 
solid, and round their jambs, provided there is no frame; 
but if there is a frame, they measure solid and half round 
their jambs. 

"4. Chimney abutments and common pillars to meas- 
ure front and both ends for length. 

"5. All walls, however thin, to be reckoned eighteen 
inches thick. 

" 6. All partition walls to be measured from out to 
out."* 

* "NOTE. The above rules are for workmanship only. When the 



260 NELSON'S COMMON-SCHOOL ARITHMETIC. 

When measuring hewn stone, neatly piled before it is 
laid in the wall, and in computing masonry for public 
works, it is customary to reckon 25 feet to the perch. 
Twenty per cent, is deducted for loose stone in piles. 

1. How many perches in a pile of hewn stone meas- 
uring 28 feet long, 5 feet high and 4 feet broad? 

28X5X4 

- --- =22^<>, or 22 per. 10 ft. 



2. What quantity of stone is in a pile 150x15x12 
feet? 

3. What will it cost to build the foundation of a house, 
which is 75 feet long, 16 feet wide and 7 feet high, wall 
18 inches thick, @ $2.25 a. perch, including materials? 

4. What will it cost to build a cellar that is 16 feet 
square, with walls 2 fe,et thick and 8 feet high, @ 82.75 
per perch, material included? 

Answers: $173.73. 1080 perches, $113.77. 

BRICKLAYERS' WORK. 

294. Bricklayers' work is computed by the thousand 
bricks. The usual dimensions of a brick are 8 inches 
long, 4 inches broad and 2 inches thick. There are 21 
bricks in a cubic foot of wall, mortar included. The gov- 
ernment standard is 1000 bricks to 40 cubic feet. A 
brick of the above dimensions weighs 4^ Ibs. 

A bushel of sand weighs 113 pouuds ; of slacked lime, 
51 pounds. 

workman furnishes materials, the value of the materials, both stone 
and mortar, in the above extra measurements, as well as the value 
of the materials saved by doors and windows, is to be deducted 
from the foot of the bill; that is, neither stone nor mortar is to be 
charged in any instance where they are not used." 



BRICKLAYERS' WORK. 261 



RULES OF MEASUREMENT OF THE BRICK- 
LAYERS OF CINCINNATI. 

"All lengths shall be exterior or taken on the outside, 
from corner to corner, and for every return or cross sec- 
tion at openings deducted, nine inches by height for work 
and materials, or one foot by height for workmanship 
shall be allowed. Autaes and Pilasters returns allowed 
in all cases. 

" All octagon or circular work of a radius of three feefc 
or less shall measure double, and for larger radius a less 
but fair allowance shall be made. All walls cut up for or 
coped with brick, shall measure one foot additional height, 
and all walls both cut up for and coped with brick, two 
feet additional height. All work with a batter or mate- 
rial deviation from plumb line, shall measure once and a 
half. 

"Flemish or plumb bond fronts shall measure solid in 
all cases, and stock or pressed brick with tuck joints shall 
measure double, with the additional cost of pressed brick 
if furnished by the contractor. 

" Kettles or stills shall measure solid, and those of 50 
feet or less, exterior surface, once and a half; and those 
of 25 feet or less, exterior surface, double. Fire fronts, 
when cased, double measurement. 

"Culverts or sewers, 9 inches thick and two feet or less 
i in diameter, and those of half-brick thick and three feet 
or less in diameter, shall measure solid. All circular 
work shall measure exterior girth. 

"All openings of sectional area greater than ten feet 
;ii shall be deducted, except those in bond fronts and ovens, 
which shall not till exceeding fifty feet; then half of such 
; | shall be deducted. 



362 NELSON'S COMMON-SCHOOL ARITHMETIC. 

"Twenty-one bricks shall be allowed to the cubic foot 
for all brick-work, and the same proportion for thicker 
or thinner walls from one brick in thickness upward; 
forty- three bricks shall be allowed per yard for brick 
paving when flat, and eighty for paving on edge in ascer- 
taining the amount of bricks only; but when paving is 
done and measured as brick-work, the sand shall be al- 
lowed for in the measurement, and all materials on edge 
or cut to, waste allowed for also. 

"Cisterns, when measured for brick-work and materials, 
half brick wall shall count six inches and whole brick wall 
ten inches, and cistern arches and those over circular 
vaults shall in all cases measure double. 

"The floor joists shall govern the height of stories in 
all cases till two stories make more than twenty-four feet 
for carrying materials, and then twenty-four feet shall be 
allowed for two stories and each ten feet of the additional 
height** a story. When new work is built upon an old 
building, or one built by another contractor, all such work 
shall be measured and allowed for an additional story in 
height for labor of carrying materials. 

"Old bricks in piles shall be subject to the usual dis- 
count in ascertaining the quantity, and in the absence of 
special agreement shall, if on the premises and sound, be 
valued same as new bricks at the kiln; if from a burnt 
building, or otherwise unsound', shall be valued by the 
measurer accordingly. 

"When hands or materials are furnished for an em- 
ployer, in the absence of agreement or contract, actual 
cost, together with fifteen per cent, thereon, shall be 
charged by the contractor. 

"Lumber and materials for scaffolding, mortar beds, etc., 
and vessels for holding water, shall, in all cases, be fur- 
nished by the employer, unless otherwise agreed upon, iu 



STONE-CUTTERS. 263 

which case a reasonable charge shall be made by the con* 
tractor. 

"For all work not embraced in, or provided for by the 
foregoing, a fair and reasonable allowance shall be made 
by the measurer." 

1. Plow many thousands of brick will be required to 
build a wall 90 feet long, 6 feet high and 20 inches thick? 

90X6=540 square feet of surface. 

In one square foot of a 20-inch wall there are 35 bricks j 
in 540 square feet there are 540x35 = 18900 bricks, which, 
divided by 1000=18.9, or 18 T 9 . 

2. In a house there are 6200 square feet of 20-inch 
wall and 2000 square feet of 12-inch wall; what will be 
the cost of building, at 11.50 a thousand, including price 
of brick and laying? 

3. What will it cost to pave a yard 25 by 50 feet, and 
a walk 75 by 5 feet, @ 50 cents a yard including mate- 
rials ? 

Answers: $90.28, 2978.50. 

4. A cistern is 8 feet in diameter and 12 feet deep (av- 
erage measure) ; what will be the cost of building at 40 
cents a barrel? 

8 2 X12X.1865=143.23 bbls., 143.23x40c.=$57.29. 

Instead of multiplying by .7854 (as required by Art. 
), and dividing by 4211 (the number of feet in a bar- 
rel), we merely use the quotient arising from .7854-7- 
4.211:=r.l865, as a multiplier. 

The true pitch of a roof is obtained by making the rafters three- 
fourths of the width of the building. 

The Gothic pitch is that produced by making the rafters as long 
as the building is wide. 

For information relating to the department of Building, we are 
indebted to R. B. Moore, Esq. 



264 NELSON'S COMMON-SCHOOL ARITHMETIC. 



STONE CUTTERS. 

AVERAGE PRICE AND RULES OF MEASUREMENT FOR COM- 
MON FREESTONE WORK. 

295. In measuring plain stone work, all the dressed 
faces of the stone are taken, and the whole reduced to su- 
perficial measurement.- For instance, a step 4 feet long, 
14 inches wide and 7 inches thick would be measured as 
follows : 

Length, 4 ft.-f 7 in. -(-7 in.=r:5 ft. 2 in. 
Width, 14 +7 =1 9, 

and 5 ft. 2 in.Xl ft. 9 in.=9 ft. 1", or 9 ft. 

REMARK. It will be observed that the ends have been measured 
twice; this is in accordance with custom. 

Window-sills are measured by the running foot, includ- 
ing the projections of the ends. Prices, for 7-inch wide, 
and 4 to 5 thick, per foot, 18 to 25 cents. 

Water-table. The stone in front of a house and on a level 
with the door-step. Measured as above, 37-^ cents per foot. 

Ashler or slab front. Face measure, adding all worked 
ends. Price, 40 cents per superficial foot.* 

Flagging. Superficial measure, 2^- inches thick, 25 
cents; 3 inches, 30 cents; 4 inches, 40 cents; 6 inches, 50 
cents; 8 inches, 60 cents. 

Fire-wall doping, running measure, 11 by 2 inches, per 
foot, 25 cents. 

Chimney coping, 2^ to 3 inches thick, per foot, 30 cents. 

Coping caps, common size, each, $2. 

Hearths, common thickness, per superficial foot, 40 cents. 

* Stone-cutters set (build) their own work, the charge for which 
is included in these prices. 



PLASTERING. 265 

Edge curlings, for walks, 2 to 3 inches thick, per linear 
foot, 25 cents. 

Door and window cornice, not exceeding 6 inches thick 
and 6 inches projection, per foot, 37-J- cents. Measured 
length and returns 3 girts from wall to wall. 

Piers for open fronts, face measure, taking the girt. 

A common door-piece, comprising 2 each, plinths, piers 
and caps, with lintel, cornice and blocking, will cost from 
$50 to $75. 

All cut stone should be laid in cement. 

Foundations and excavations for steps should be sunk 
at least three feet below the surface, else the action of the 
frost on the earth will be liable to displace the stone. 

Mortar should not be exposed to the action of frost 
until it is set. 

During the heats of summer, mortar is injured by a too 
rapid drying in the wall; to prevent this, the other ma- 
terials, stone or brick, should be thoroughly moistened 
before being laid; and afterward, if the weather is very 
hot, the masonry should be kept wet until the mortar 
gives indications of setting. In very warm weather, the 
top course should always be well moistened by the work- 
men on quitting their work for any short period. 

PLASTERING. 

296. The business of the plasterer is to cover brick 
and stone work, ceilings and partitions, with plaster, and 
prepare them for paper, paint, etc. ; also, to form cornices 
and such other decorative portions of walls and ceilings as 
may be executed in plaster or cement. 

To lay off a square corner or a right angle, with a carpenter's rule: 
Measure 3 feet from the corner in one direction, 4 feet in another 
direction and separate the points 5 feet apart. 

23 



266 NELSON'S COMMON-SCHOOL ARITHMETIC. 



"CINCINNATI PLASTERERS' RULES OF MEAS- 
UREMENT. 

" 1. All work shall be measured superficially, including 
openings. All heights shall be taken from the floor to the 
ceiling. 

U 2. All staircases eight feet wide and under shall be 
measured double; all over eight feet, once and a half. 

"3. All passages four feet wide and under shall be 
measured once and a half; all over four feet, once and a 
fourth. 

"4. All inclined ceilings to measure once and a half. 

"5. All dormor windows, closets and privies to be meas- 
ured double. 

" 6. All octagon and circular work, except ceilings of 
rooms, to be measured double. All arched ceilings of 
rooms to be measured once and a half. 

"7. The deductions for openings occasioned by doors 
and windows, when the workman furnishes materials, shall 
be, for lathwork, one-eighth; for brick walls, one-fourth. 

"8. The materials for scaffolding and mortar-beds and 
vessels for holding water, are, in all cases, to be furnished 
by the employer." 

STUCCO WORK. 

Moldings, cornices not over 12 inches girth, are meas- 
ured by the running or linear foot. Flowers^ sometimes 
singly; in moldings, per superficial foot. 

Eighteen laths will cover a yard; 500 laths, a square 
of 100 feet. 

Estimate work is made from a bill of prices, the carpenter agree- 
ing to work for a certain percentage on the bill. 



PAINTING, PAPER-HANGING AND GLAZING 267 

1. "What will be the cost of plastering a room 18 feet 
long and 16 feet wide, with a ceiling 10 feet high, at 18 
cents? 

18 
J.6 

34x^68, length round the room. 
68x10=680 square feet in walls. 
18X16=288 " " " ceiling. 

9)968 
107.55 yds. at 18c.=$19.359, or $19.36. 

2. How many square yards of plastering in a hall 84 
feet long and 40 feet wide, with a ceiling 18 feet high, 
having a, space of 600 square feet occupied by windows? 

Ans. 802f yards. 

PAINTING, PAPER-HANGING AND GLAZING. 

297. Painters' Work is measured by the square yard, 
and charged for according to the number of coats, the 
quality of the paint and the description of the work. 
Sash frames are charged for singly or by the piece, and 
sashes by the number of squares. Lettering is charged 

.for by the lineal foot. Common lettering, 25 cents; gild- 
ing, 75 cents. 

Painting is sometimes charged for by the quantity of 
paint used, and the time spent in putting it on. 

The calculations being so simple, it is considered un- 
necessary to give any examples. 

298. Glazing is sometimes charged by the square foot, 
and sometimes so much per light. When estimated by 
the foot, it is usual to include the sash in the measure- 
ment. 

299. Paper-hanging is charged for by the piece. The 
commoner qualities measure about 7f yards by 19 inches, 



268 NELSON'S COMMON-SCHOOL ARITHMETIC. 

or about 35 square feet; the better qualities, say from 50 
cents a piece, are 9 yards by 21 inches, or 47 square feet. 
Border paper is made in rolls of the same dimensions as 
the wall paper, each roll or piece containing two or more 
strips, each of which is called a piece, and is sold at about 
the same price as the paper it is designed to match. 

Dealers in paper usually contract for the hanging, and 
charge from 20 to 25 cents a piece, according to quality. 
300. To find the quantity of paper required for a 
room, compute the number of square feet in the walls; 
deduct the openings and divide the result by the number 
of feet in a piece. The space occupied by the base will 
allow for waste and matching pattern. 

1. A room is 16 feet square and has a ceiling 12 feet 
high, with two doors 7 feet by 4, two windows 7 by 3 
feet and a fire-place 4 feet square; how much paper will 
be required to hang it, and what will be the whole cost, 
including hanging paper 25, border 35 cents? 

16 X 4=64, length of wall around the room. 

64X12=768 square feet of wall 
Windows, 42 feet. 
Doors, 56 " @ 

Fire-place, 16 " 

114 

654 square feet to be covered. 

654 

=19 pieces. 

35 number of feet in a piece. 

19X25 cents=cost of paper 4.75 

Length of border, 64 feet, or 3 pieces at 35 1.05 

Hanging 22 pieces @ 20 cents,* 4.40 

Total cost $10.20 

* The border is included in the 22 pieces. 



FARMING. 269 



GAS FITTING AND PLUMBING. 

301. Gas fitting is charged for per foot of pipe, vary- 
ing according to size. Fittings and chandeliers, per piece. 

302. Plumbing is charged for like gas fitting. For 
ordinary house work, say from 25 to 30 cents per foot of 
pipe ; sheeting by weight. 

FARMING. 

The young farmer will find it to his interest to be a 
good arithmetician. For those who have not had the ad- 
vantages of an early education, we will introduce a few of 
the simpler and more necessary calculations, suggesting, 
at the same time, that during his leisure moments the 
farmer should master the entire science as contained in 
this little work, which any person of ordinary ability, who 
can read and write, may accomplish without the aid of a 
teacher. 

303. To find the number of acres in a field or tract 
of land having four square corners,* we multiply the 
length by the breadth, and divide the result by 160, if 
the measure was taken in rods; or by 43560, if taken in 
feet.f 

1. The length of a field is 125 rods and its breadth 112 
rods; how many acres are in it? 

112X1^5=14000, which, divided by 160, gives 87J. 

* A figure having square corners, and all its sides equal, is a 
equare; one having its opposite sides equal, a rectangle or parallelo- 
gram. 

tin a square rod there are 272|- square feet. When there are feet 
remaining to be reduced to rods, it will be sufficiently accurate to 
divide by 272. 



270 NELSON'S COMMON-SCHOOL ARITHMETIC. 

2. A lot of land is 400 feet long by 110 feet broad; 
how many acres does it contain? Ans. 1 acre 1.7 rods. 

304. To lay off a given quantity of land. 

3. What should be the length of a strip of land 30 rods 
broad to contain 6 acres? 

In 6 acres there are 960 rods, which, divided by 30= 
32 rods. 

305. To find the contents of a field in the shape of ct 
right-angled triangle, we multiply the two shorter sides to- 
gether, and take one-half the product. 

REASON. A right-angled triangle is half a square or parallelo- 
gram, formed by drawing a line between opposite corners. 

4. The shorter sides of a right-angled triangle are 45 
and 60; required the contents. Am. 1350. 

306. To find the quantity of grain or coal in a bin or 
wagon, we multiply the length, breadth and height to- 
gether, and for grain divide the product by 1.2444,* if 
the diminsions are given in feet; or by 2150.42, f if given 
in inches. For coal, by 1.555, or 2688. 

5. A wagon is 8 feet long, 5 feet broad and 18 inches 
deep; how many bushels of corn does it contain? 

8X^X1^=60, the number of cubic feet. 
1.2444)60.0000(48.21, or 48 bushels.* 
49776 

102240 NOTE. Two ciphers were annexed to the 

99552 dividend to correspond with the decimals of 
the divisor, and produce the ivhole numbers, 48, 
and two more ciphers were annexed to pro- 
duce the decimal .23, or - 2 75 L or i. 

19920 

6. How many bushels of grain in a bin measuring 4 
feet every way? Ans. 51^, nearly. 

*Feet in a bushel. t Inches in a bushel. 



FARMING. 271 

307. To find flie quantity of wood or bark in a pile, we 
multiply the three sides given in feet, as before, and di- 
vide by 128, the number of feet in a cord. 

7. How many cords of wood in a pile 40 feet long, 7 
feet high and 4 feet broad? Ans. 8|- cords. 

SOS. Having two sides and the contents of a box, to find 
the third side, we divide the cubical contents by the pro- 
duct of the two sides. 

REASON. Since the product of the three sides equals the con- 
tents, the contents divided by two of the sides will give the third 
side. 

8. A box is 2 feet wide and 3 feet high; how long 
should it be to hold 25 bushels of coal ? 

In 25 bushels there are 2688X25 or 67200 cubic inches. 
In 2 ft. there are 24 inches. 

" 3 " " 36 " 24x36=864=area of the end. 
672-5-864==77|5|, or 6 ft. 5f in. 

9. What must be the height of a bin that will hold 300 
bushels of wheat, if its- length is 30 feet and its width 4 
feet? Ans. 3 ft. 1J in. 

10. What must be the depth of a box 16 inches square 
to hold a bushel? a box 10 inches square to hold a peck? 
one 8 inches square to hold half a peck? 

To find the side of a cube that will hold a certain quan- 
tity. See Cube Root. 

309. To find the quantity of grain when heaped against 
a wall or partition, take half the perpendicular height for 
one side, and multiply it by the length and breadth, as in 
Art. 306. 

310. To find the number of cubic feet in a round log. 
See Art. on the Cylinder. To find the solidity of a cyl- 
inder, wa multiply the area of the end by the length. 



272 NELSON'S COMMON-SCHOOL ARITHMETIC. 

11. How many feet are in a log 12 feet long and 30 
inches in diameter? 

In 30 inches there are 2J, or 2.5 feet: 2.5X2.5X-7S54 
=4.9087, the area of the end. 4.9087X12=58.9044, or 
58 T 9 Q feet, the solid contents. 

REMARK. This method of calculating, though correct, is seldom 
used for practical purposes. It is customary for lumber merchants 
to throw off one-third of the diameter, and consider the remainder 
the side of a square log. A log of the dimensions named in the 
preceding question would thus measure only 33J feet, or one-third 
of 100 feet; and is thereby taken as the standard of measurement 
in some of the Western States. See Lumber Business. 

311. Trade, or barter. 

12. How many cords of wood, at S3. 75 a cord, should 
I get for 50 bushels of wheat at $1.12J a bushel? 

50x1-12^=^56.25, which, divided by $3.75, will give 
the number of cords. 5625-i-375 15 cords. 
PROOF. 15 cords at $3.75=856.25. 

13. How many pounds of sugar, at 8 cents a pound, 
should I get for 127 pounds of buttor, at 12J cents a 
pound? Ans. 198. 

14. How many days' work of a man, at 75 cents a day, 
will be equal to 45 days' work of a man at $1.25? 

Ans. 75. 

15. How many cords of wood, at $2.25, will be equal 
to 150 cords, at $3.50? 

16. How many yards of muslin, at 8 cents a yard, can 
be bought for 5 dozen chickens, at $1.25, and 15 dozen 
eggs, at 8J- cents? 

LUMBER BUSINESS. 

312. Lumber measure comprises solid and superficial 
measure. Round logs are measured by deducting one-third 



LUMBER MEASURE. 273 

of the diameter for waste, and calling the remainder the 
side of a square log. 

1. To find the contents of a round log 24 inches in di* 
ameter and 30 feet in length. 

SOLUTION. Deducting J from 24 for waste, we have 16, 
which, squared=256 inches, and multiplied by the length^ 
640 feet board measure. 

In some places only J is deducted for pine lumber.* 

Planks or joists are sometimes reckoned by face meas- 
ure; that is, the dimensions of one side of the board are 
taken instead of the solid contents. A 16-foot board 2 
inches thick by 12 inches broad would measure 32 feet 
board measure, or 16 feet face measure. 

In some places, the saw-log is taken as a standard of 
measurement for round timber. A log 12 feet long and 
30 inches in diameter is the standard in some parts of the 
west. In Pennsylvania, a saw-log is one that will cut into 
200 feet of lumber. 

313. To measure timber partly squared, it is customary 

to deduct the "wane" (the length of the corner) from the 

thickness of the log, and call the remainder one side. A 

log 18 inches thick, with a "wane" 3 inches, would be 

called one of 18 by 15 inches. 

2. In an octagonal log, 25 feet long 20 inches thick, 
with a wane 4 inches, how many solid feet are there? 

Ans. 55|. 

3. There are 150 logs, the average length and breadth 
of which are 20 feet by 22 inches, wane 3 inches ; required 
the number of solid feet they contain. Ans. 8708J. 

* Inch measure is taken as the standard for lumber. If a board 
is under an inch, it is measured as a full inch; and if over an inch, 
it is reduced to inch measurement. A plank 2 inches thick would 
be considered as two boards 1 inch thick. 



274 NELSON'S COMMON-SCHOOL ARITHMETIC. 

4. In a raft there are 450 boards 16 feet long and 1^ 
inches thick, and measuring in the aggregate 757 feet 
broad; how many feet of lumber (board measure) does it 
contain? How many face measure? 

Ans. 18168 board measure, 12112 face measure. 

5. How much lumber can be cut from a tree measuring 
20 feet long and 14 inches diameter at the smaller end, 
allowing for waste one-fourth of the diameter? 

HOUSEKEEPING. 

314. Housekeepers, and ladies generally, ought to be 
familiar with the operations in arithmetic which apply in 
computing house rent, servants' wages, board bills, inter- 
est, the quantity of carpet to cover a floor or paper for a 
room, etc. 

HOUSE RENT. 

Landlords, in renting by the year, usually collect their 
rents quarterly ; but when renting monthly, collect monthly. 

By a quarter is meant three calendar months. As, for 
instance, if a house is rented on the 17th of April, the 
quarter would expire on the 17th of July. 

When a house is rented for a year, the tenant is liable 
for the rent during the whole of that time unless the 
landlord accepts another in his stead. A verbal lease for 
a year is binding. A lease for three or more years should 
be recorded. 

Tenancy begins on obtaining possession. When there 
is a lease, however, and the time not stated, it is pre- 
sumed to commence on the date of the instrument. 

When the tenant does not remove at the end of the 
year, or two weeks afterward, he will be regarded as hav- 
ing rented for another year. 

Interest can be collected on rent from the day it is due. 



HOUSEKEEPING. 275 

A tenant is released when the landlord accepts a sub- 
stitute. 

A married" woman can not make a lease or take one in 
her own name. 

A tenant at will is liable for rent as long as he occu- 
pies the premises. 

For interest calculations, see page 143. 

1. A house which rents at $75 a month is occupied 
from January 3 to February 9 ; what is the amount of 
rent? 

Rent for 1 month= 75 , 

" " 6dsor$= 15 

$90 

2. Required the rent for a house from April 3 to Au- 
gust 5, at $1000 a year. 

3 mo= 1 1000 EXPLANATION. From April 8 to July 3 

~7^:T is 3 months, or 1 fourth of a year; 1 fourth 

of $1000 gives $250. From July 3 to Au- 
gust 5 is 1 month and 2 days; 30 days is 
1 third of 3 months, and the rent for that 
338.888 time is $83.333, and 2 days is 1 fifteenth 
or $338.89 of a month, giving the rent for that time, 
$5.555. 

3. Required the rent for a house from December 1 to 
January 12, at $50 a month. Ans. $68.33. 

4. What will be the rent of a house from January 20 
to August 9, at $750 a year, payable quarterly? 

Ans. $416.67. 

The Teacher can give more of such questions as he finds it nec- 
essary. 

SERVANTS' WAGES. 

Servants are hired by the week or month of four weeks 
or calendar month, and are entitled to wages every day, 
Sunday included. 



30 = 



276 NELSON'S COMMON-SCHOOL ARITHMETIC. 

* 

5. A girl hires on September 3 and leaves on October 

9; what will be her wages at $3 a week? 

From September 3 to October 9 is 36 days, or 5 weeks 
1 day. 

Wages for 5 weeks at $3= 15.00 
Wages for I day=\ of $3= .428 

15.428 
or $15.43 

6. A man is hired on June 9 at $40 a calendar month, 
and is discharged on September 3; what is the amount of 
his wages? Ans. $113.33. 

This is computed in the same way as house rent. 

7. What will be the wages of a man for 7 months and 
7 days at $33 a month? Ans. $238.70. 

To find the quantity of carpet for a floor. 

Most carpeting is made one yard in breadth. Brussels 
and velvet carpeting is usually made only f of a yard or 
27 inches, though sometimes it is made |- and even f, or 
double the usual breadth. 

Oil cloths vary in breadth from 3 to 24 feet, as follows : 
3 ft. 9 in., 4 ft. 6 in., 7 ft. 6 in., 12 ft., 18 ft. and 24 ft. 

Matting is of three kinds: China Cocoa, Manilla and 
Cane. China matting is made of a kind of rushes. It- 
looks neat but does not wear long. The best kinds are 
Gowqua and Manning. 

Cocoa matting is made of a kind of grass. The best 
quality is called "diamond A," from the brand found 
upon it. 

Common ingrain carpeting may be matched by cutting 
through the center of the pattern ; but expensive carpets 
can be matched only by persons experienced in the busi 
ness. Some of them require two webs, others more, to 






HOUSEKEEPING. 277 

make a pattern. Carpet dealers usually furnish their car- 
pets made to any dimensions, and even lay them when re- 
quired. 

The quantity of carpet required for a room is found by 
multiplying the length by the breadth, in feet or inches, 
and dividing by the number of square feet or inches in a 
yard. For f carpet, divide the square feet by 6-J; for 
yard, divide by 9.* 

ANOTHER WAY 

Is to find the number of Ireadths required, and multiply 
it by the length of the room. 

8. How much ingrain carpet will be required to cover 
a room 15 by 20 feet? 

Cutting the carpet in its greatest length, there would 
be 5 breadths, which, multiplied by the length, gives 100 
feet, or 33J yards. 

Required the quantity of ingrain carpet to cover three 
rooms, measuring as follows: one room 12 by 16 feet; 
one, 16 by 21; one, 15 by 19; and one room, 20 by 25 
with velvet carpet. 

9. What quantity of velvet carpet will cover a saloon 
20 by 40 feet? 

The breadth, 20 feet, reduced to inches 240, which, 
divided by 27 inches=9 breadths, nearly. The length, 
40 feet, multiplied by 9^360 feet, or 120 yards. 

for calculations pertaining to wall paper, see Paper 
Hanging, page 267. 

For calculations pertaining to shopping, see page 111. 

For weights and measures, see Tables. 

* The spaces for fire-places, etc., will allow sufficient for waste. 
Carpets should be cut a few inches short to allow for stretching. 



278 NELSON'S COMMON-SCHOOL ARITHMETIC. 

To find the cost of articles sold by the dozen. 

1 article will cost y 1 ^- of the cost of a dozen. 

2 articles " " " " " " " 

g u it it l u u it it it 

4 u it a ^_ it it it it it 

5 " " " 5 times the twelfth. 

6 u " " | of the cost of a dozen. 

8 " " " 2 times the third. 

9 " " " 3 " " fourth. 

10 " " " $ off. 

11 " " " -J. " 



XXXVII. GEOMETRICAL DEFINITIONS. 

An Angle is the opening between two lines that meet 
in a point. 

A Right Angle is made by one straight line standing 
perpendicular to another. 

An Obtuse Angle is wider than a right angle. 

An Acute Angle is less than a right angle. 

A Triangle is a figure having three sides and three 
angles. 

An Equilateral Triangle has all its sides equal. 

An Isoscetes Triangle has two of its sides equal. 

A Scalene Triangle has all its sides unequal. 

A Right-Angled Triangle has one right angle. 

An Obtuse-Angled Triangle has one obtuse angle. 

An Acute-Angled Triangle has all its angles acute. 

A Quadrangle or Quadrilateral is a four-sided figure, 
and may be 

A Parallellogram, having its opposite sides parallel ; 

A Rectangle, having four right angles, sides unequal; 



GEOMETRICAL DEFINITIONS. 279 

A Square, having all its sides* equal, and its angles 
right angles; 

A Rhombus or Lozenge, having its sides equal and no 
right angle; 

A Rhomboid, a parallelogram, with no right angles ; 

A Trapezium, having unequal sides; 

A Trapezoid, having only two sides parallel. 

Polygon, a plain figure having more than four sides. 

A Pentagon has nre sides, a hexagon six, a heptagon 
seven, an octagon eight, a nonegon nine, a decagon ten, etc. 

A Circle is a plain figure, bounded by a curved line, all 
points of which are equidistant from the center. 

An Arc is any part of a circumference. 

A Chord is a straight line joining the extremities of an 
arc. 

A Segment of a circle is a part of a circle bounded by 
an arc and its chord. 

The Radius of a circle is a line extending from the 
center to the circumference. 

A Quadrant is a quarter, a sextant a sixth of a circle. 

A Zone, a part of a circle included between two parallel 
chords. 

A Prism is a solid, the sides of which are parallelo- 
grams. It may have three or more sides. 

A Pyramid is a solid with regular sides, tapering to a 
point. 

A Cylinder is a solid of uniform thickness, having its 
ends circular. 

A Cone is a round pyramid, having a circle for its base. 

The Circumference of a circle is the line by which it is 
bounded. The Diameter is a line drawn through the cen- 
ter and terminating at the circumference. The Radius is 
a line drawn from the center to any point in the circum- 
ference. 



280 



NELSON'S COMMON-SCHOOL ARITHMETIC. 







j 








"1 

, j. 






I 

j 



XXXVIII. MENSURATION. 

315. To find the area of a square, multiply the length 
by the breadth. 

1. Let one side of the annexed parallelo- 
gram be 3, and the other 4, then the area 
will be 3X4=12. 

2. What is the area of a square, the side 
of which is 3 feet 6 inches. Ans. 12J. 

316. To find the area of a Rhombus or Rhomboid, we 
multiply the length by the perpendicu- 
lar height. The reason for this will be 

evident from an inspection of the fig- 
ures. The triangle A C m of the rhom- 
bus applied to the side B D, will make 
a square ; and the triangle of the rhom- 
boid applied to the other side will make 
a parallelogram. 

3. Let the perpendicular height (A 
m) be 20, and the length (A B or C D) 
be 30, then the area will be 20x30600. 

4. What will be the area of a rhomboid, the perpen- 
dicular and length being 15 and 25? Ans. 375. 

317. To find the area of a right-angled triangle, we 
multiply the perpendicular by half B 
the base, or the base by half the per- 
pendicular. 

REASON. The part A m n, applied to the j 
part B C m n, with the line A m applied to m 

B, and the point A on the point B, will make a parallelogram, with 
a base half of that of the triangle. 




MENSURATION. 281 

5. Let the base be 10 and the perpendicular 8; then 
1X 8=40, the area. 

6. The perpendicular is 16 and the base 120; what is 
the area? Ans. 960. 

318. When the triangle is not right-angular, half the 
base multiplied on the height will 

give the area. 

REASON. The triangle C A D is half the 
rhomboid. 

7. Let the base be 30 and the height 20, then 20 X- 3 / 
300, the area. 

319. When the perpendicular is not given, the area can 
be found by subtracting each side from half of the sum of 
the sides; then by multiplying these three remainders and 
half the sum of the sides together, and extracting the 
square root of the product 




8. Let the sides be 5, 7 and 10 ; then - - --- ^ 



11 7=4 

1110=1 6X4X11=264, the sq. root of which is 16.3. 

9. What is the area of a triangle, the sides of which are 
50, 30 and 40? Ans. 600. 

320. To find the area of a circle, multiply the square 
of the diameter by .7854. 

Multiply half the circumference by half the diameter. 

321. To find the side of a square equal in area to a given 
circle, multiply the circumference by .2820948, or the di- 
ameter by .8862269. 

10. The diameter of a circle is 100; required the side 
of a square having the same area. 

.8862269X100=88.62269, side of a square, 
24 



282 



NELSON'S COMMON-SCHOOL ARITHMETIC. 




322. To find the side of an inscribed square, multiply 
the diameter by .7071068. 

11. The diameter of a tree is 2 feet; re- 
quired the side of a square log that may 
be cut from it? 

.7071068X2=1.4142136, or 1 foot 5 
inches nearly. 

323. Having the side of a square, to find the diameter of 
a circumscribed circle, multiply the side by 1.4142136. 

324. From the side of a square to find the circumference 
of a circumscribed circle, multiply the side by 4.4428934. 

325. To find the diameter, multiply the side by 
1.1283791. 

326. From the side of a square, to find the circumference 
of a circle of equal area, multiply the side by 3.5449076. 

327. To find the area of a trapezium, we divide it into 
two triangles, and the sum of the areas 

will be the area required. 

12. Let the diagonal A C be 100, 
and the perpendiculars B m and D n : 
30 and 35 ; then 

(35+30) X 100 

V ^ ' =3250, the area. 

REMARK. The areas of irregular polygons are found by dividing 
the figures into triangles, and taking the sum of their areas. 

328. To find the circumference of a circle, we multiply 
the diameter by 3.1416, or 3, because 

the circumference of a circle is 3^ times 
greater than the diameter. 

13. Let the diameter be 5; then 3.1416 
X 5=15. 708, the circumference. 

The diameter is found by dividing the circumference 
by 3.1416. 





MENSURATION. 



283 



329. To find the area of a regular polygon, add all the 
sides together, and multiply by the perpendicular drawn 
from the center of the polygon to the middle of one of its 
sides; or, 

Multiply the square of the side of the polygon by the 
number standing opposite to the number of its sides in 
the following table: 



3 


0.4333012 


8 


4.8284271 


4 


1. 


9 


6.1818242 


5 


1.7204774 


10 


7.6942088 


6 


2.5980762 


11 


9.3656404 


7 


3.6339124 


12 


11.1961524 



14. The side of a pentagon (a five-sided figure) is 20 
yds. and its perpendicular 13.76382; required the area. 

20X5X13.76382 
First method, =688.191, Ans. 

Second method, 20 2 X 1.720477=688.19, Ans. 

15. The side of a nonegon is 50 inches; required its 
area. Ans. 15454.5605. 

330. To measure the heights of objects, the tops of which 
can not be reached, the shadow cast by the tree may be 
used. Measure the length of the shadow cast by the ob- 
ject, and that of some object the length of which is 
known; then the shadow of the known object will be to 
that of the first as the length of the known object to 
the length of the first 

16. Let the known object be a man, who, without his 
hat, measures (in his shoes) 5 feet 8 inches, and whose 
shadow measures 15 feet, while the shadow of a tree meas- 
ures 120 feet. 

15 ; 120 : : 68 inches : the height of the tree in inches. 

120X68 

=544 inches, or 45J feet. 



284 NELSON'S COMMON-SCHOOL ARITHMETIC. 

MEASUEEMENT OF SOLIDS. 

331. . To find the solidity of a cube, multiply the side 
by itself, and the product again by the side. 

17. The side of a cubical block of marble is 5 feet 7 
inches; what is the solid contents? 

5_7^x5 T 7 .TX5 T 7 2 =174 feet, nearly. 

332. To find the solidity of a parallelopipedon (a solid 
figure with square corners), multiply the length, breadth 
and thickness together. 

18. A log measures 7 feet in length and 15 by 20 
inches in thickness; required the solid contents. 

7X15X20=2100 square inches, or 14 T 7 ^ feet. 

333. To find the solidity of a prism, multiply the area 
of the end by the length. 

19. What is the solidity of a prism whose ends are 
equilateral triangles, each side of which is 4 feet and 
height 8 feet? 

An equilateral triangle is made up of two right-angled 
triangles, the perpendicular of which is 
found by taking the square root of the 
difference of the squares of half the base 
and the other side. 




y / 4 2 2 2 =3.464, the perpendicular, 
m n. 

3.464x2=6.928=area of the end. 

6.928X8=55.424 feet, the solidity. 

334. To find the solidity of a cone, multiply the area 
of the base by one-third the height. 

20. A cone is 10 feet in diameter and 10 feet high; re- 
quired the solidity. 

10 2 X -7854=78.54, area of the base, which, multiplied 
by J the height, 3J=261.8, the solidity. 



MENSURATION. 285 

21. How many cubic feet in a cone whose diameter is 12 
feet, and its perpendicular height 100? Ans. 3769.92 ft. 

335. To find the solidity of a pyramid, multiply the 
area of the base by one-third the height. 

22. A square pyramid has a base of 4 feet and height 
of 12 feet; required the solidity. 

4x4 area of base. 
16X^=64 for the solidity. 

23. The spire of a church is an octagonal pyramid, each 
side at the base being 5 feet 10 inches, and its perpen- 
dicular height 45 feet; also each side of the cavity or 
hollow part at the base is 4 feet 11 inches, and its per- 
pendicular height 41 feet ; how many solid yards of stone 
does the spire contain? Ans. 32|-, nearly. 

336. To find the solidity of the frustrum of a cone or 
pyramid* 

Find the sum of the areas of the two ends, and of a 
geometrical mean between them, and multiply by one- 
third the perpendicular height. 

24. What is the solid contents of a frustrum of a 
square pyramid, whose sides are 5 and 3, and perpen- 
dicular height 12? 

5 2 X .7854=19.6350 area base. 
32 X .7854= 7.0686 area upper end. . 
1 /r9.635x7.0686=11.7810 geometrical mean. 
38.4846 

4 one- third the height. 
153.9384 

25. What is the solidity of a squared piece of timber, 
its length being 18 feet, and sides of the bases 18 and 12 
inches? Ans. 28.5 ft. 

*A segment is a piece cut off by a plane, parallel to the base; a 
fruslrwn is what remains at the base. 



286 NELSON'S COMMON-SCHOOL ARITHMETIC. 

26. How many cubic feet of timber in a tapering log 
14.25 ft. long, diameters 9 and 18 in.? Am. 14.689 ft. 

Comparison between the globe, cylinder and cone, the di- 
ameter and heights being 100: 

Solid contents of the cylinder, 785.4 
" " " sphere, 523.6=f of the cylinder. 

" " " cone, 261.8=4 u " 

REMARKS. 1. The cone cut out of a solid cylinder, whose diame- 
ter and height are equal, will leave a part equal to the solidity of 
a sphere of the same diameter. 

2. A square pyramid, whose height and side are equal to the side 
of a cube, if cut out of the latter, will leave f of the cube. 

GAUGING. 

The process of finding the capacity of barrels, etc., is 
called gauging. 

337. llfiving the head and Lung diameter and the di- 
ameter between them, to find the capacity of a barrel or 
cask in gallons, we add together the square of the head 
and bung diameters and twice the middle diameter, and 
multiply the sun; by the length, and that by .0004721 for 
imperial gallon. 

27. A cask, having for head, bung and middle diameter 
30, 36 and 33, and length 40 inches, holds how many im- 
perial gallons? 

30 2 -f36 2 -f(33x2)X40x. 0004721=42.72 galls. 

Practical method for measuring small cylindrical vessels, 
is to multiply the square of the diameter by 34, and that 
by the height in inches, and point off four figures. The 
result will be the capacity in gallons. 

28. An oil-can measures 12 inches in diameter and 2 
feet in height; required the contents in gallons. 

12 2 X34X 24=117504=11.75 or 11$ galls. 



THE METRIC SYSTEM. 287 



XXXIX. THE METRIC SYSTEM. 

338. The Metric System is a decimal system of weights 
and measures, of French origin, deriving its name from 
Meter, the unit of measure upon which the system is based. 

Since 1840 it has been adopted by most European gov- 
ernments, including that of Great Britain in 1864, and 
has been in use by men of science every-where. During 
its last session (39th), Congress authorized its use in this 
country, and made provision for its immediate introduc- 
tion into post-offices.* 

339. The units of measure are the meter, are, liter, and 
stere; and the unit of weight the gram. Other denomina- 
tions are formed from these by prefixing Greek or Latin 
numerals; the former for denominations above the unit, 
and the latter for denominations below the unit. The 
Greek prefixes are deka, 10; hecto, 100; kilo, 1000; and 
myria, 10000. A dekameter is ten times and a hectome- 
ter one hundred times the length of a meter. The Latin 

*A Bill to authorize the use of the Metric System of Weights and 

Measures. 

J3e it enacted by the Senate and House of Representatives of the 
United States of America, in Congress assembled, That from and after 
the passage of this act, it shall be lawful throughout the United 
States of America to employ the weights and measures of the 
Metric System; and no contract, or dealing, or pleading in any 
court, shall be deemed invalid or liable to objection because the 
weights or measures expressed or referred to therein are weights 
or measures of the Metric System. 



288 NELSON'S COMMON-SCHOOL ARITHMETIC. 

prefixes are deci, T Vth; C entL T J-g-th; milli, y^Vu*^- A 
decimeter is one-tenth and a centimeter one-hundredth of 
the length of a meter. 



LIST OF NAMES AND THEIR PRONUNCIATION. 



Name. 


Pi'onunciation. 


Abbreviation. 


Meter, 


Me'ter, 


M. 


Kilometer, 


Kil'o-meter, 


K. M, 


Hectometer, 


Hec'to-meter, 


H. M, 


Dekameter, 


Dek'a-meter, 


D. M, 


Decimeter, 


Des'i-meter, 


d. m. 


Centimeter, 


Cent'i-ineter, 


c. m. 


Are, 


Aer, 


A. 


Hectare, 


Hector, 


H, A. 


Centare. 


Seut'iir, 


c. a. 


Stere, 


Stere, 


S. 


Dekastere, 


Deckl-stere, 


D. S. 


Decistere, 


Des'i-stere, 


d, s. 


Gram, 


Gram, 


G. 


Dekagram, 


Dek x a-gram, 


D. G. 


Kilogram, 


KiFo-gram, 


K. G. 


Hectogram, 


Hec^o-gram, 


H. G. 


Decigram, 


Des^-gram, 


d.g. 


Centigram, 


Sent^-gram, 


e.g. 


Milligram, 


Mil'li-gram, 


m. g. 


Tonneau, 


Ton x no, 


Ton. 


Millier, 


MiKli-er, 


Mil. 


Liter, 


L^ter, 


L. 


Kiloliter, 


KiKo-k-ter, 


K. L. 


Hectoliter, 


Ilect/o-le-ter, 


H. L. 


Dekaliter, 


Dek x a-te-ter, 


D. L. 


Deciliter, 


Des / i-le-ter, 


d. 1. 


Centiliter, 


SenKi-le-ter, 


c. 1. 


Milliliter, 


MiKli-lg-ter, 


m. 1. 


NOTE. Denominations 


below units arc 


abbreviated with small 


letters. 







THE METRIC SYSTEM. 289 



MEASURES OF LENGTH. 

840. Besides its being the base of the new system, the 
Meter is the unit of measure for lengths, and is one ten- 
millionth part of the distance from the Equator to the poles, 
and is equivalent to 39.37 inches ordinary measure. 



Myriameter 10,000 Meters or 393685 inches. 

Kilometer 1,000 " or 39368.5 " 

Hectometer = 100 " or 3936.85 " 

Dekameter 10 < or 393.685 " 

Meter = 1 or 39.3685 " 

Decimeter = T ^th " or 3.9368 " 

Centimeter == ^th " or .39368 " 

Millimeter T?r Wth " or .03936 



REMARK. Meters, when combined with lower denominations^ 
are treated as whole numbers, and the latter as decimals. 

The denominations most used are the Kilometer, Centimeter^ 
and Millimeter. 7 Myriameters, 8 Kilometers, and 6 Hectome* 
ters would be written 78.5 K. M. 

SQUARE OR SURFACE MEASURE. 

341. The unit of measure for large surfaces is the Are, 
from which are derived the Hectare and Centare. For 
smaller surfaces the denominations are the same as for 
measures of length, with the addition of the word square. 

Centare = 1 sq. meter, or 1560 sq. inches. 

Are = 100 sq. meters, or 1 sq. dekameter, or 119.6 sq. yds. 

Hectare = 10,000 sq. meters, or 1 sq. hectometer, or 2.471 acres. 

1 sq. decimeter, d. w., 2 = T J^ sq. meter 

1 sq. centimeter, c. m., 2 =-. T J^ sq. decimeter, or T Q JQQ sq. meter. 

1 sq. millimeter, m. m., 2 = T J^ sq. centimeter, or T ^^J^^^ sq. c. m. 

27.354 M. 2 = 27 sq. meters, 35 sq. decimeters, 40 sq. centimeters. 



290 NELSON'S COMMON-SCHOOL ARITHMETIC. 



CUBIC OR SOLID MEASURE. 

342. The Stere may be called the unit for cubic meas- 
ure. It is equal to a cubic meter or 1.308 yards. 

1 dekastere, D. S.,=W steres, or 13.08 yards. 

1 decistere, d. s., T V stere, or 0.1308 yards. 

1 cubic decimeter, d. m.f = T ^Vtf cubic meter. 

1 cubic centimeter, c. m.^= T ^ inf cubic decimeter, or -j-^J^^ M. 

1 cubic millimeter, m.wi.,= T ^ cubic centimeter, or 



The Stere, etc., is used for measuring fire-wood and 
lumber, and, for computing large numbers, is preferred 
to the other denominations. 

MEASURES OF CAPACITY. 

343. The Liter is the unit of measure for capacity, 
and is equal to a cubic decimeter or 1.0567 quarts of 
United States liquid measure. 

TABLE. 

Kiloliter = 1,000 liters, or 1 cubic meter of water. 
Hectoliter = 100 " or ^ cubic meter. 
Dekaliter = 10 " or 10 cubic decimeters. 
Liter of water weighs 1 kilogram. 
Deciliter = j 1 ^ liter, or ^ cubic decimeter. 
Centiliter = T ^ liter, or 10 cubic centimeters. 

A milliliter of water weighs a gram. The liter and 
hectoliter are most in use. 

WEIGHTS. , 

344. The Gram is the unit of weight, and is equal to 
15.432 grains Troy, which is the weight of a cubic centi- 
meter of pure water at its greatest density. 



THE METRIC SYSTEM. 



291 



Kilogram = 1,000 grams. 
Hectogram = 100 " 
Dekagram = 10 " 
Gram 1 gram. 

Decigram = " 



Centigram = ^ gram. 
Quintal = 100 kilograms. 
Tonneau 1,000 kilograms, 
or 2,204 Ibs. 



REMARK. The gram and its subdivisions are used in com- 
pounding medicines, and wherever great accuracy is required. 

The kilogram is the denomination most used, and weighs a 
little over 2 pounds. The quintal and tonneau are used for 
heavy weights, but may be expressed in kilograms. 



COMPARISON OF METRIC DENOMINATIONS WITH THOSE IN 
PRESENT USE. 

MEASURES OF LENGTH. 



NAMES. 


VALUES. 


EQUIVALENTS 

IN USE. 


Mvriameter .. 


10 000 meters 


6.2137 miles. 


Kilometer 


1,0(10 ik 


0.62137 " 




100 " 


328 l-l2th feet. 


Deka meter 


10 " 


393.7 inches. 






39 37 " 


Decimeter 
Centimeter 
Millimeter 


l-10th of a meter 
1 -100th of a meter 
1-loooth of a meter 


3.937 " 
0.3937 " 
0.0394 ** 



MEASURES OF SURFACE. 



NAMKS. 


VALUES. 


EQUIVALENTS IN USE. 


Hectare 10,000 


square meter.* 


2.471 acres. 


A re 100 






Con tare 1 


ki meter 


l">f>0 square indies. 



MEASURES OF CAPACITY. 



NAMES. 


No. of 
Liters. 


CUBIC MEASURE. 


DRY MEASURE. 


LIQUID OR 
WINK ME AS. 


Kiloliter or Stere. 
Hectoliter 


1,000 
100 


cubic met-r 
-10th cubic meter 


1.3()<S cubic yards... 
2 bus & 3 3 r > pecks 


264. 17 grails. 
26.417 alls. 


Dekaliter 


10 




9 og quart* 


2 6417 galls 


Liter 


1 


cubic decimeter 


'MIX quart 


I u r >67 quarts. 


Deciliter 
Centiliter 
Milliliter 


1-10 
1-100- 
1-1000 


-1<> cubic decimeter. 
cubic centimeters 
cubic centimeter... 


6. 1022 cubic inches. 
0.6102 cubic inch ... 
0.06102 cubic inch.. 


O.S45 trill. 
0.:W8 fluid oz. 
0.27 fluid drin. 



292 NELSON'S COMMON-SCHOOL ARITHMETIC. 



WEIGHTS. 



NAMES. 


No. OF 
GRAMS. 


WEIGHT IN QUANTITY OF 
WATER AT MAXIMUM 
DENSITY. 


EQUIVALENTS 

IN USE. 


Millinr or Toiine.au... 


i 000 <H)0 


1 cubic meter 


2204 6 pounds Av. 


Quintal 


1(1(1 Off! I 




220.46 " 




10 (too 




22.046 " 




1,(MXJ 


1 liter 


2.2046 *' 


Hectogram 


100 


1 deciliter . 


3.5274 ounces 


Dekagram 


10 


10 cubic centimeters 


3527 ounce 




1 


1 cubic centimeter 


15.'432 grains T oy. 




1-10 




1.5432 " 


Centigram 
Milligram 


1-100 
1-llXJO 


10 cubic millimeters 
1 cubic millimeter 


0.1543 grain 
0.0154 " 



CALCULATIONS PERTAINING TO THE METRIC SYSTEM. 

The principal recommendation of the new system is the 
simplicity of its arithmetic. Under it the long and in- 
tricate calculations of the present system are unknown. 
Denominations are changed from one to another by the 
use of the decimal point, with sometimes one or more 
ciphers, while addition and other operations are per- 
formed precisely as in whole numbers and decimals. 

1. In 35.2 D. M. how many M? Ans. 352 M. 
Solution. In 1 D. M. there are 10 M., in 35.2 D. M. 

there are 35.2 times 10 or 352 M. 

2. In 35142 millimeters how many meters? 

Ans. 35.142 M. 

Solution. 1000 m. m. make 1 M., hence we divide by 
1000, which gives 35.142 M. 

3. In 31475 M. how many kilometers? 

Ans. 31.475 K. M! 

4. Reduce 371.2 D. M. to meters. Ans. 3712 M. 

5. lleduce 45.67 M. to decimeters. Ans. 456.7 d. c. 

6. lleduce 13456 decimeters to meters. Ans. 1345.6 M. 

7. In 213.21 M. M. how many dekameters? 

Ans. 213210 D. M. 

8. lleduce 3157.2 c. m. to meters. Ans. 31.572 M. 



THE METRIC SYSTEM. 293 

9. A man traveled 80.5 K. M. in a day, how far would 
he travel in 5 days at the same rate ? Ans. 152.5 K. M. 

10. A ship steams 15 K. M. in an hour, how long would 
it take her to go a distance of 50 myriameters ? 

Ans. 33^ hours. 

11. At 2 cents per kilometer, what would be the cost 
of traveling 351.27 M. M. ? Ans. $70.25. 

12. A man walked a distance of 351.27 K. M. in 9 days, 
how many was that per day? Ans. 39.03 K. M. 

13. In 253 d. m., 7 c. m., 5 m. m., how many meters ? 

Ans. 25.375 M. 

14. In 157 d. m 2 ., 35 c. m 2 ., how many square meters ? 

Ans. 1.5735 M 2 . 

15. At 15 cents per M 2 ., what would 25 H. M 2 . of lum- 
ber cost? Ans. $375. 

16. How many square meters in a floor which measures 
6 M. long and 5.25 M. broad? Ans. 31.5 M 2 . 

17. A pavement measures 39.2 M. long and 4.15 M. 
broad, how many square M. does it contain? 

Ans. 162.68 M. a 

18. A strip of land 120 M. long contains 342 ares, 
what is its breadth? Ans. 285 M. 

19. What will 545.37 centares cost at $4.15 per are ? 

Ans. $22.63. 

20. From 1374.27 M 2 . take 7.14 ares. 

Ans. 6.6027 M 2 . 

21. At $50.50 per H. A., what will 1371.154 ares cost? 

Ans. $692.43. 

22. In 37.27 M 3 . how many decisteres? How many 
dekasteres? Ans. 372.7 d. s., 3.727 D. S. 

23. At $1.50 per M 3 ., what will 157.28 dekasteres cost? 

Ans. $2359.2. 



294 NELSON'S COMMON-SCHOOL ARITHMETIC. 

24. What will 153 S. of wood cost at $1.75 per S.? 

Ans. $267.75. 

25. $875.55 was paid for 413.5 S. of wood, what was 
the cost per stere? Ans. $2.17. 

26. A cistern is 3 M. long, 2 M. wide, and 1 M. deep, 
what weight of water will it contain? 

Ans. 13227.6 Ibs., or 6 tonneau. 

27. What will a decistere of wood cost, at $3.57 per 
stere? Ans. 36 cents. 

28. Find the price of 53 liters of wine, at $3.50 per 
dekaliter. Ans. $18.55. 

29. A vessel measures 53.5 liters, what weight of water 
does it contain? Ans. 53.5 kilograms. 

30. In .3514 H. L. how many liters? Ans. 35.14 L. 

31. In 145.37 M 3 . how many liters? Ans. 145370 L. 

32. Find the price of 5.3 liters of wine, at $1.75. 

Ans. $92.75: 

33. At 57 cents a meter, what will 317.5 meters cost? 

Ans. $180.975. 

34. An are of land cost $53, what was the price per 
meter? Ans. 53 cents. 

35. A kilogram costs $37.50, what will a dekagram 
cost? Ans. 37-J- cents. 

36. How many ares in a square piece of land which 
measures 150 meters in length and 500 in breadth? 

Ans. 750. 

37. At $25 a kilogram, what will a dekagram cost? 

Ans. 25 cents. 

38. At $7.25 a stere, what will 320 steres of wood cost? 

Ans. $2320. 



THE METRIC SYSTEM. 295 

343. To reduce the denominations of the ordinary to 
those of the Metric System. 

1. In 3 miles 35 rods, how many meters? 

3 miles 35 rods=197010 inches, which, divided by 39.37, 
(the number of inches in a meter,) gives 5004.06 meters. 

2. In 5^ yards how many meters? Ans. 4.877. 

3. Reduce 37 |~ feet to meters. Ans. 11.43. 

4. In 29 inches how many centimeters? Ans. 73.66. 

5. In 6 inches how many millimeters? Aus. 152.4. 
61 How many square meters in 57 rods? 

Ans. 1441.68. 

7. In 5 acres how many ares? Ans. 202.34. 

8. How many liters of wine in 37 gallons? 

Ans. 140.06. 

9. In 57 yards of carpet how many meters? 

Ans. 52.12. 

10. How many liters in 5^- bushels? Ans. 193.82. 

11. How many cubic meters in 3759 cubic feet? 

Ans. 106.44 

12. Reduce 3 tons to tonneaus. Ans. 2.72. 

13. At $2.50 a liter, what will 20 gallons cost? 

Ans. $189.27 

14. At 75 cents a meter, what will 135 yards cost? 

Ans. $92.58. 

344. To reduce denominations of the Metric to those 
of the ordinary system. 

1. In 175 meters how many yards? 

Ans. 191 yds., 1 ft., If in. 

In 1 meter there are 39.37 inches, in 175 meters there 
are 175 times as many. 39.37X175=6889.75 inches, 
which, reduced to yards, rr=191 yards, 1 foot, ly^ inches. 

2. In 379.53 meters how many yards? Ans. 415.058. 



296 NELSON'S COMMON-SCHOOL ARITHMETIC. 

3. Reduce 743.5 K. M. to miles. Ans. 461.988 

4. In 2435 millimeters how many inches? 

Ans. 95.866.. 

5. Reduce 50 pounds to kilograms. Ans. 22.68. 

6. At $2.75 per kilogram, what will 375 Ibs. cost? 

Ans. $467.77, 

7. At 35 cents per lb., what is it per kilogram? 

Ans. 77 cents. 

8. At $10.50 per cord, what is it per stere? 

Ans. $2.90. 

9. At $1.25 a bushel, what should wheat be sold at 
per liter? Ans. 3^ cents. 

10. A bushel of oats weighs 33 pounds, what should 
it weigh per liter? Ans. 1.14 Ibs. 

11. How long will it take a man to travel a distance of 
57 K. M. who walks at the rate of 3^ miles per hour? 

Ans. 10.119 hours. 

The following multipliers, taken from a pamphlet writ- 
ten by Prof. H. A. Newton, of Yale College, will facilitate 
the labor of converting the denominations of one system 
into those of the other. They will be found sufficiently 
correct for ordinary purposes : 

MetersX39.3685=inches. InchesX.0254=nneters. 
MetersX3.2807=feet. FeetX-30481 meters. 
MetersXl-09357=yards. YardsX.91444=meters. 
MetersX-19883r=:rods. RodsX5.0294=meters. 
KilometersX-62135=miles. MilesXl-6094=kilometers. 
Sq. metersX!550 sq. inches. Sq. inchesX.0006452=sq. M. 
Sq. metersX10.763=rrsq. feet. Sq. feetX-09291 sq. meters. 
Sq. metersXl-196 sq. yards. Sq. 3'ardsX.8362=sq. meters. 
AresX3.953=sq. rods. Sq. rodsX.2529^ares. 
HectaresX2.4709=racres. AcresX.4047=hectares. 



THE METRIC SYSTEM. 297 

Hect.aresX-003861=sq. miles. Sq. milesX259.=rhectares. 
LitersX33.81 fluid ounces. Fluid ouncesX-02958 liters. 
LitersXl-05656=quarts. QuartsX-9465r=liters. 
LitersX.26414:=gallons. GallonsX3.786=liters. 
HectolitersX2.837=bushels. BushelsX.3524=hectoliters. 
LitersX61.012 cubic inches. Cubic inchesX.01639=rliters. 
HectolitersX3.531=cubic feet. Cubic feetX.2832=rhectoliters. 
SteresXl-3078 cubic yards. Cubic yardsX-7646 steres. 
SteresX-2759=cords. CordsX3.625z=Steres. 
GramsXl5-44 grains. GrainsX-0648=grams. 
KilogramsX32.147--=troy ounces. Troy oz.X-03108=rK. G. 
KilogramsX35.30=avoirdupois oz. Av. oz.X-02833^K. G. 
KilogramsX2.681=Troy pounds. Troy poundsX-373=K. G. 
KilogramsX2.206=Av. pounds. Av. poundsX4536=K. G. 
TonneausX-985=long tons. Long tonsXl-015=:tonneaus. 
TonneausXl-103=short tons. Short tonsX-9066 tonneaus. 

1. In 37 meters how many yards? 

Ans. 37X1.09357=40.462 yds. 

2. In 40.462 yards how many meters? 

Ans. 40.462 X. 91444=37 meters. 

3. In 120 kilograms how many pounds Avoirdupois? 

Ans. 120X2.206=264.72 pounds. 

4. In 26.472 pounds how many kilograms? 

Ans. 264.72X. 4536=120. kilograms. 

The meter is nearly 3 feet 3 inches and 3 eighths, or 
3 feet 3J- inches. 

REMARKS. The word Meter means measure, as in gas- 
meter, or gas-measure. 

Stere means solid, as in stereotype, solid type, or rather 
a solid mass of type. Stereoscope, an instrument to make 
two pictures look like a solid. 

Are signifies area, a surface included within given lines. 

Quintal is from the Latin root centum, a hundred ; for- 
merly a hundred weight. 



PAGE. 

Vocabulary of Technical Terms used in Business 5 

Tables of Money, Weights and Measures 15 

Notation and Numeration 26 

Addition...! 33 

Subtraction 43 

Multiplication 49 

Division 60 

Short Division 61 

Percentage 68 

Long Division 72 

Properties of Numbers 77 

Multiplication by Aliquots 79 

Easy Fractions 82 

The Mercantile Profession 94 

Bills and Invoices 100 

Compound Numbers 117 

Short Methods 125 

Marking Goods Gain and Loss 132 

Commission and Brokerage 139 

Interest and forms of notes, etc 142 

Simple Interest ; 143 

Banking 154 

Checks, Drafts, etc 163 

Receipts 165 

Time Tables for Averaging.. 168 

Discounting Notes % 170 

True Discount 174 

Compound Interest 178 

Average Equation of Payments , 180 

(298) 



NELSON'S COMMON-SCHOOL ARITHMETIC. 299 

Accounts Current - 194 

Exchange 196 

Fractions. * 202 

Extinction of Fractions 204 

Decimals 206 

Multiplication of Decimals v ... 210 

Addition of Decimals 208 

Subtraction of Decimals 209 

Division of Decimals 211 

Reduction of Decimals 213 

Multiplication of Common Fractions 216 

Subtraction of Common Fractions 222 

Division of Common Fractions 219 

Addition of Common Fractions... 223 

Ratio and Proportion 227 

Compound Proportion 231 

Partnership 233 

Joint Stock Companies, etc 235 

Importing 239 

Foreign Exchange 240 

Insurance .-. 244 

Duodecimals 246 

Involution and Evolution 248 

The Trades, Farming, etc 256 

. Useful Hints on Building 257 

Geometrical Definitions 278 

Mensuration 280 

Gauging 286 

The Metric System, 287 



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